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ALGEBRA FOR BEGINNERS 


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Wes: 
BY ae 
DAVID EUGENE SALITH, Px.D. 
PROFESSOR OF M EM NICS IN TEACHERS COLLEGE 
COLUMBI NIVERSITY, NEW YORK 


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GINN. & COMPANY 


BOSTON - NEW YORK - CHICAGO » LONDON 


' ENTERED AT STATIONERS’ HALL 


COPYRIGHT, 1904, 1905 


By DAVID EUGENE SMITH 
ALL RIGHTS RESERVED 


35.10 


The Atheneum Press 


GINN & COMPANY > PRO- 
PRIETORS + BOSTON: U.S.A. 


PREFACKH 


This book is intended, as the title indicates, for pupils 
beginning the study of algebra. There is a growing dispo- 
sition to introduce this subject somewhat earlier than was 
formerly the case, and with this has come a demand for a 
simple, interesting, and sufficiently scientific text-book for 
beginners. Such a text-book should show the utility of 
algebra, should form a connecting link between arithmetic 
and the more scientific works to be studied later, and 
should stimulate a desire to proceed further in mathe- 
matics. It is to meet this demand in the spirit described 
that this book has been prepared. 

The time for introducing the work depends upon circum- 
stances. In some cases elementary algebra of this nature 
should be begun in the first year of the high school, while 
other conditions make it advisable to take it up in the 
latter part of the grammar-school course. In either event 
it is desirable that pupils should have some knowledge of 
algebra before they leave school. For those who are not 
to pursue the subject further this book furnishes such 
algebra as is necessary for the intelligent reading of form- 
ulas and the solution of equations found in elementary 
industrial manuals. Those who continue their school work 
will find the subject treated in this book in such a way as to 
stimulate an interest in their later work, and will meet no 
obsolete forms that must be unlearned before proceeding. 

In sequence of topics the author has continued the plan 
adopted in his arithmetics, that of recognizing the value 
of the various courses of study in use in different parts 
of the country. Modern curricula no longer sanction for 

lll 


4 9OeK 


492K 
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Lore PREFACE 


beginners the plan of treating each topic but once. On 
the contrary, they suggest the repetition of the most 
important portions of algebra, although favoring a some- 
what exhaustive treatment of each subject whenever it is 
under discussion. Of the three chapters of this book, the 
second covers some of the ground of the first, and the third 
reviews some of the topics treated in the second. ‘The first 
two chapters furnish sufficient work for schools that devote 
part of a year to algebra and part to arithmetic. The third 
chapter may be used if a full year is given to the subject. 

The work seeks to interest the pupil in the subject at 
once by showing him its utilities. The formula which the 
artisan meets in his trade journals and the equation which 
throws so much hght upon business arithmetic find place 
in the early pages. With these applications is combined 
the recreation element, as seen for example in the finding 
of numbers which satisfy given conditions, — an element 
which lends much interest to mathematics. 

Oral algebra, like oral arithmetic, 1s necessary to lead 
to rapidity and to an understanding of general processes. 
Hence enough types have been suggested to form a basis 
for the best of all oral work, that which comes sponta- 
neously from the teacher and the class. 

While a large number of genuine applications have been 
made in the domain of the pupil’s present and prospective 
experiences, scientific and financial problems in which he 
has no interest have been omitted. With the applications 
has gone a large number of those abstract, formal prob- 
lems so necessary for drill in rapid algebraic work. ‘These 
“problems without content” have an interest in them- 
selves, and give to the elementary pupil some of that 
pleasure which comes to the more advanced student in 
the discovery of positive truth in the domain of pure 


science. 
DAVID EUGENE SMITH. 


CONTENTS 


CHAPTER I 


THE USES OF ALGEBRA; THE OPERATIONS WITH INTEGERS 
AND FRACTIONS ; THE EQUATION 


PAGE 

Some OF THE USES OF ALGEBRA 1 
ForMvULAsS . : : . ; ‘ : : : : 2 
EQuaTIONS : d : : : ‘ : : : ; 4 
LETTERS IN SOLVING PROBLEMS . : ‘ ; > : 6 
Tue USE OF & ; : F : , : . : 4 9 
SoME OF THE TERMS USED IN ALGEBRA 4 : ; : 16 
Soms Usges or Monomiats. , : : : ‘ 5 AN 
Some Uses oF POLYNOMIALS . : : ; ‘ : Pat 
THe NeGaATIvE NuMBER . é 4 : : : : Oy y- 
CurvE TRACING . ‘ : : : : : 2 ; 25 
ADDITION . . , A : d ; : , : a As) 
SUBTRACTION . : : : : ‘ : f ; , 34 
How To USE PARENTHESES s 4 : ; 2 ; . 40 
MULTIPLICATION. P ° : : ; : : . 42 
DIvIsION. : ; : 4 ‘ : ‘ . : og Bb 
Factors AND MULTIPLES : . ; : - : , 50 
Facrors : : : ; ; ; , : : fe 60 
MULTIPLES . 53 
FRACTIONS ; 55 
Uses OF FRACTIONS . : ; : ; as ; : 57 
REDUCTION OF FRACTIONS é : ; ; : A = OS 
IMPROPER FRACTIONS . 5 ; : ; ; : : 61 
FRACTIONAL EQUATIONS . : F : 4 : 2 Oe 
ADDITION OF FRACTIONS ; ; , A ; A ; 65 
SUBTRACTION OF FRACTIONS ; : . d : 5 rea 8 
MULTIPLICATION OF FRACTIONS : : : ae ae : 68 
DIvisION OF FRACTIONS. : ’ ; ; , : a uae 
LINEAR EQUATIONS ‘ i ‘ . , ‘ é : 74 


vi CONTENTS 


CHAPTER II 
OPERATIONS CONTINUED; FACTORING; PROPORTION ; 
EQUATIONS 

PAGE 
MULTIPLICATION 5 ; , ; : E ; : Geo 
FacTORING . : ; : ; ; : ; : ; 88 
Division. : : F . f : 4 ; : we OF 
FRACTIONS : ; ? : , : : : : : 99 
REDUCTION OF FRACTIONS ( : d : : : be OO 
ADDITION OF FRACTIONS . : 3 : : : : 106 
SUBTRACTION OF FRACTIONS . : : : : ; . 108 
MULTIPLICATION OF FRACTIONS . : é : ; ; 110 
Division Or FRACTIONS . : ; i : . 4 ae |) 
EQUATIONS INVOLVING FRACTIONS . ; : : ’ q 112 
PROPORTION : ; : ; : , : : : ragltg G2) 
SquaRE Root : : : ‘ : ‘ : , : 129 
QUADRATIC EQUATIONS ; : ; : ; ; : -| Loo 


CHAPTER III 


FRACTIONS CONTINUED ; ROOTS ; SIMULTANEOUS EQUA- 
TIONS ; THE COMPLETE QUADRATIC 


FRACTIONS ; : : 4 ; ? ‘ , : e186 
FRACTIONAL EQUATIONS. , , 2 : f : . 140 
SIMULTANEOUS EQUATIONS ; : : : ; 2 : 143 


QuapDRATIC EQuaTIons : : : i ; : : +. “163 


ALGEBRA FOR BEGINNERS 


CHAPTER I 


THE USES OF ALGEBRA; THE OPERATIONS WITH INTEGERS 
AND FRACTIONS; THE EQUATION 


SOME OF THE USES OF ALGEBRA 


1. Numbers represented by letters. — In arithmetic we often 
represented numbers by letters. We learned that 

If one thing costs d@d dollars, 5 things will cost 5 x d 
dollars, which we write $5d; and n things will cost $ nd. 

2. How we indicate multiplication. —In algebra the absence 
of a sign indicates multiplication. 

It is not so in arithmetic, for 51 means 50+1; but in 
algebra ab means a x 0. 


ORAL EXERCISE 


1. If a rectangle is 12 ft. long and 7 ft. wide, what is its 
area? Ifitis 7 ft. long and w ft. wide, what is its area? 

2. If a train travels at the rate of 30 mi. an hour, how 
far will it travel in 10hr.? If it travels m miles an hour, 
how far will it travel in 2 hours? 

3. While the hour hand of a clock passes over 5 1-min. 
spaces, how many does the minute hand pass over? While 
the hour hand passes over ” spaces, how many does the 


minute hand pass over? 
1 


ko 


USES OF ALGEBRA 


3. Rules stated by letters.— We have just seen that the 
area of a rectangle / long and w wide is Jw. If 7 and w are 
numbers of feet, 7w is the number of square feet in the 
rectangle ; if inches, dw is the number of square inches. 
If a represents the area, then the statement 

a = lw 
is a simple rule for finding the area of a rectangle. 

4. Formulas. — <A rule stated in letters is called a formula. 
For example, you may have found in arithmetic that the 
circumference of a circle equals the radius multiplied by 
twice the number 3.1416, 3.1416 being represented in 
mathematics by the Greek letter 7 (pi). But it is much 
easier to express this rule by the formula 


c= 277. 


ORAL EXERCISE 


1. If a triangle has a base 4 and height 6, what is the 
area? Suppose it has a base 6 and height h? 

2. Given a=1bh, find the value of @ when )=7 and 
h=6; whend=6andA=7; when b=h=1O, 

3. If an automobile has a constant velocity of 8 miles 
an hour, how far will it go in 8 hours? If it has a 
constant velocity of v miles an hour, how far will it 
go in ¢ hours? 

4. From Ex. 3, what meaning do you get from the state- 
ment d=vt? (Think of d as standing for distance.) 
What is the value of d when v=15, t=}? | 

5. If 5 men can do a piece of work in 8 days, how long 
will it take 4 men, working at the same rate? If m men 
can do it in d days, how long will it take a men? Read 
from the formula a rule for solving all such examples. 


FORMULAS 5) 


6. If c=277, find the value of ¢ when r= 5, w having 
the value stated in § 4. 


7. If nis an integer, does 2 7 represent an even number 
or an odd one? What kind of a number does 2 » + 1 rep- 
resent ? What is the value of each ifn = 5? 


8. Represent a number divisible by 2 (see Ex. 7); not 
divisible by 2; divisible by 3; divisible by 5. 

9. If a man saves $5 a week, what will he save in 
8 weeks? If he saves d dollars a week, what will he 
save in w weeks? 


10. If m represents any-number, how shall we represent 
twice the number? 4% of the number? the number plus 
5% of itself? the number minus 10% of itself? 


11. What is the area of a parallelogram whose base is 
6 in. and height 4in.? Read from the formula a = bA the 
rule for finding the area of a parallelogram of any given 
base 6 and any given height h. 


12. What is the interest on $200 for 2 years at 5%? 
Read from the formula i= prt the rule for finding the 
interest on any principal p, at any rate rv, for any time ¢ 
expressed in years. 

13. What is the volume of a box 8 in. long, 5 in. wide, 
and 3 in. deep? Read from the formula v = lit the rule 
for finding the volume of a rectangular solid, given the 
length, breadth, and thickness. 

14. A man rows at the rate of 5 mi. an hour in still 
water. How far can he go downstream in an hour, the 
stream flowing 3 mi. an hour? If he rows 7 miles an hour 
and the stream flows s miles an hour, how far will he go? 
Read from the formula d=7—=s a rule for finding the 
distance he will be able to row upstream. 


4 USES OF ALGEBRA 


ORAL EXERCISE 


1. How many 1-lb. weights will just balance 32 oz. of 
sugar in these scales? 


2. Suppose I take half as much sugar, what about the 
weights? Suppose I add 8 oz. to the sugar? Suppose I 
take away 8 oz. of sugar? 
Suppose I double it? 


3. If the weights on 
| = both sides of the scales 
i Nae 7 just balance, what can we 
= 46 say about the balance if 
we multiply them by the 
same number? divide them by the same number? add 
equal weights to both sides? subtract equal weights from 
both sides ? 


4. Suppose I have equal sums of money in two banks, 
and wish to keep equal sums there. If I put $10 more 
in one bank, what else must I do to keep the equality? 
Suppose I take $a from one bank? Suppose I double 
the amount in one bank? Suppose I take out half of the 
amount in one bank? 
























































5. The equation. — An expression of equality between two 
quantities is called an equation. 


For example, z +5 =7. Here we see that x = 2. 


6. The principles of the equation. — We have found above 
that the following principles are true: 
1. If equals are added to equals, the results are equal. 
2. If equals are subtructed from equals, the results are equal. 
3. If equals are multiplied by equals, the results are equal. 
4. If equals are divided by equals, the results are equal. 
| 


EQUATIONS D 


ORAL EXERCISE 


1. What must be added to 3 to make 8? to 8 —5 to 
make 8? as x — 5 to make x? 

2. Tee 5 = 20, what must be added to these equals to 
give the yalue of x? What is the value of «? 

3. If «+5 = 30, what must be subtracted from these 
equals to give the value of x? What is the value of x? 

4. If 3a =7, what does x equal? By what did you 
eialtiply: these equals to find the value of x? 

5. If bf = = 15, what does # equal? By what did you 
divide these equals to find the cee of a? 


7. Illustrative problem. — If 2a eat 9, what is the 
value of a? 


Felt wo +l = 9: 
2. Then 2a = 8, by subtracting Il from these equals. 
3. Then a = 4, by dividing these equals by 2. 


WRITTEN EXERCISE 


Find the value of the letter in each of the equations in 
Hizs. 1-12: 


i 5 = 24. 2 = 36. 

gaa — 40: Ay On 45, 

eee — 17. 6. dn = 164. 

7. 83m = 415. Shae Sa bey 

90 a+ 73 = 122. 10 2 
11. m — 63 = 149. 12. x — 87 = 236. 


13. What number added to 173 equals 361? 
14. What number is that of which } equals 237 ? 


6 USES OF ALGEBRA 


8. Letters used in solving problems. — One of the chief 
uses of letters in algebra is in solving problems. A prob- 
lem in arithmetic can often be more clearly solved in this 
way than by numbers alone. 

For example, I am thinking of a number. When it is 
multiplied by 2, and 7 is added to the result, the sum is 33. 
What is the number ? 


Solution using a letter: 


1. If I am thinking of n, then 
2n is twice the number, 
2n + 7 is this added to 7, 


and 2n + 7 = 33, as stated in the problem. 
2. Then 2n = 26, by subtracting 7 from equals. 
3. n = 13, by dividing these equals by 2. 


Check or Proof. 2x 138 +7 = 38. 


Solution without using letters: 
Because twice the number added to 7 equals 33, therefore if 
7 be taken away from 33 there will remain twice the number. 
Therefore 26 is twice the number. Therefore once the number 


is half of 26, or 13. 


The solutions compared: 


2n+7 = 33. 33 
Subtracting 7, 7 
BirenoG: 2)26 

Therefore n= 138. 13 


9. We therefore see that the two solutions are the same, 
but that the letters make the reasoning clearer. Therefore 

1NWrite a letter for the number sought. 

2. Use this letter in the statement of the problem. 

3. This will give an equation, as shown above. 

4. Solve this equation. 


EQUATIONS T 


ORAL EXERCISE 


Iam thinking of a number. When it is multiplied by 3, 
and 7 1s added to the product, the result is 40. What is 
the number ? 

1. By what letter do you wish to represent the number? 

2. Then how shall we represent 3 times the number ? 

3. How shall we represent this added to 7 ? 

4. How shall we express this sum as equal to 40? 

5. What shall we subtract from these equals to leave 
37 on one side of the equation? 

6. What shall we do to these equals so as to leave n 
alone on one side of the equation? 

7. How shall we check or prove that our result is correct? 


As the answers to the above are given, the equations should be 
written on the board. 


WRITTEN EXERCISE 


1. I am thinking of a number. When it is multiplied 
by 7, and 7 is added to the product, the result is 7 times 7. 
What is the number? (7+ 7 = how many?) 

2. I am thinking of a number such that twice the num- 
ber and 3 times the number together equal 35. What is 
the number? (2n+37 = how many?) 

3. If to 3 times a certain number I add 138, the sum is 
43. Whatis the number? (32+ how many = 43?) 

4. If to 24 times a certain number I add 9}, the sum is 
17. Whatisthe number? (2)”-+ 9} = how many?) 

5. If I add 2} times a certain number to 3} times the 
same number, the sum is 46. What is the number? 

6. If from 69 I subtract 2, the result is 17 more than 5 
times a certain number. Required the number. 


8 USES OF ALGEBRA 


ORAL EXERCISE 


1. I am thinking of anumber. If it is multiplied by 3, 
and 1 is added to the product, the result is 7. What is 
the number? 


The teacher or pupil should write the work on the board. Pupils 
soon come to visualize work like 3n+1=7, 8n=6, n=2. 


2. If a certain number is multiplied by 5, and 4 is added 
to the product, the result is 19. What is the number? 

3. If a certain number is multiplied by 7, and 4 is added 
to the product, the result is 25. What is the number? 

4. If to twice a certain number I add 3 times the same 
number, I have how many times that number? If this 
sum is 35, what is the number? 


WRITTEN EXERCISE 


1. Ralph’s father is 39 years old, and this is 3 times 
Ralph’s age. How old is Ralph? (8 = 939.) 

2. Rob is a year more than 38 times_as old as his sister. 
He is 13 years old. How old is his sister? (8s+4+1 =?) 

3. Tom has 10 less than twice as many marbles as 
Frank. Tom has 20. How many has Frank ? 

4. Jennie’s mother is 6 years more than twice as old as 
Jennie. Her mother is 34 years old. How old is Jennie? 

5. Our club played with the Crescents. Our score was 2 
more than twice theirs. Ours was 14. What was theirs ? 

6. Our room is 33 ft. long, and this is 1 ft. more than 
twice the width. How wide isthe room? (2w+1= ?) 

7. There are 128 girls in our school, and this number is 
2 more than 6 times the number of boys in this class. 
How many boys are there in this class? (60+ 2 =?) 


EQUATIONS 9 


10. The use of x. — While we may represent a number by 


the initial letter n, or a number of dollars by d, or a number 
of marbles by m, or by any other letters, it 7s customary to 
represent by the letter x a number which ts to be found. 


Teachers will naturally encourage pupils to use other letters occa- 


sionally, particularly initial letters, where they add to clearness. But 
in general the convention of algebra should be recognized. 


ORAL EXERCISE 


Find the value of x in the following: 


owt _> 


. 2a = 201Y 2. 5x = 60. 3. Tx = 84. 

mpl eon: gr Zo t= 120. 6. 125% = 250. 

. 82+5=26. (What does 3xequal? What does « equal?) 
Meo 108 69. oe lie 41. 10, 6% +. 9 = 15, 
11. 


@tceao= lis 12. $¢-+-4=—60.. 13. 94-10 = 109: 


WRITTEN EXERCISE 


Find the value of x in the following: 


1. 12¢%4 3= 1365. 2. lla+3=1385. 
3. 102 4+ 2 — 162." 4.10”%+12= 162. 
Be oe dl 10. 6. 312+ 12 = 260. 
toe == 140, 8. 17x +11 = 300. 
9. 42%74+4= 130. 10. 512+ 45 = 300. 
11. 10+ 652 = 140. 12. 20 + 30% = 470. 


Make up problems to fit the following, writing them out 
like those on page 8; then solve: 


15. 52 +16 — 56: 16. 22+ 3a = 265. 


10 USES OF ALGEBRA 


11. Equations involving subtraction. — If from 3 times the 
number of which I am thinking I subtract 7, the remainder 
is 29. What is the number? 


1. I am thinking of a number which I call z. 


2. Then Bat = 29; 
3. Then 3x2 = 86, by adding 7 to these equals. 
4, And x = 12, by dividing these equals by 3. 


Check. 3 x 12 — 7 = 29. 


ORAL EXERCISE 


1. What number less 2 equals 6? 

2. If from a certain number I subtract D, the result is 
45. Whatis the number? , | ‘ 

3. If from 4 times a certain number I subtract 4, the 
result is 20. What is the number? 

Such problems, like all oral drill, have their greatest value when 
given extempore by the teacher. Rapidly given, with small numbers, 


the work is not difficult, and it is interesting. A little daily drill of 
this kind soon makes pupils very ready in equation work. 


WRITTEN EXERCISE 

1. If to 25 times a certain number I add 42, the result is 
667. What is the number? 

2. If to 125 times a certain number I add 50, the result 
is 800. What is the number? 

3. If from 25 times a certain number I subtract 42, the 
result is 583. What is the number? 

4. If from 75 times a certain number I subtract 25, the 
result is 200. What is the number? 

5. If from 37 times a certain number I subtract 33, the 
result is 300. What is the number? 


EQUATIONS 11 


12. Equations involving per cents. — After deducting 10% 
from the marked price of some goods, a dealer sold them 
for $13.50. What was the marked price? 


1. Let x represent the number of dollars of marked price. 
(We need not, then, trouble ourselves to write the sign $ each 
time, as we should if x represented merely the price.) 

2. Then x = 10% = 135-50; 


or D0 se = 15.00: 
because any number less 7; of itself is 7% of itself. 
3. Therefore x = 13.50 + .90, by dividing equals by .90, 
alts 


4. Therefore the marked price was $15. 
Check. $15 — 10% of $15 = $15 — $1.50 = $13.50. 


WRITTEN EXERCISE 


‘1. What number less 10% of itself equals 72? 
2. What number less 17% of itself equals 166 ? 


_ 3. Jack now weighs 84 lbs., which is 12% more than he 
weighed a year ago. How much did he weigh then? 


4. A certain school gained 15% this year over the number 
last year. It now has 161 pupils. How many had it last 
yeager tl low = 161.) 

5. A dealer saved $1968 this year from his store. This 
is 18% less than he saved last year. How much did he 
save last year? 


6. A dealer was obliged to sell some damaged furniture 
at 10% less than cost. He sold it for $85.50. How much 
did it cost? How much did he lose? 

7. A village having a population of 2040 at the last 
census found that it had lost 15% from the number at the 


preceding census. How many had it before? How many 
had it lost? 


13. 


3. Lhen 


USES OF ALGEBRA 


ORAL EXERCISE 


Find the value of x in each of the following: 


ee we Oates 
LT a 
cite aed be mer ge Be 
.2e@+ 121. 
Pee ain 59, 
we oe 16. 
oO + 44 = 25. 
de 3 = 56. 


2. 
4. 


16. 


x—3=18. 
xe —12 = 40, 
. e—14 = 60. 
, 2e—1=A49, 
2 =O Ok: 
Oe =O =i 
AO { ==00: 

lil« —3= 380. 


Illustrative problem.— Find the value of x that makes 
4-+--17 2 = 140. 


Lelia lia — 140; 
2. Then 17 «= 1386, by subtracting 4 from these equals. 


x =1386 +17 = 8, by dividing these equals by 17. 
Check. 17 X 8+4=186+4=140. 


WRITTEN EXERCISE 


Find the value of x in each of the following: 


L 
On ee awl LOT. 

. 1.06” = 424, 

WASa Seb Ree 2713 

. 152% +47 = 962. 

. 025 + 14a = 367. 

. 411 E11 ae 774. 

. 231+ 23a = 392. 

. 125% 4+ 125 = 1000. 


O FF ot WwW 


Zia = 42. 


2. 


Ol tea ie 
 GOi7 = (D9: 
. 60% = 144,90, 
WILT 2 19s 287. 


19n = 97189. 
Six — $= 1520. 

. 419 x — 88 = 800. 

, 125% — 125 = 750. 
, 1012 — 111 = 1000. 


FORMULAS 13 


ORAL EXERCISE 


1. At 5% a year, how much is the interest on $200 for 
one year? for two years? for 7 years? 

2. At r% a year, how much is the interest on $p for 
one year? for ¢ years? If the rate of interest is r, and 
the principal is p, and the number of years is ¢, what is 
the interest ? 

3. At 10% discount, how much is the discount on $50 
worth of goods? At r% discount, how much is the dis- 
count on $» worth of goods ? 


14. Further use of formulas. — We have just seen that if 
1= interest, r=rate, ¢=time (in years), and p= the 


principal, ; 
4= rp. 


WRITTEN EXERCISE 


1. If i= trp, find the value of « when ¢=6, r=49%, 
p = $300; when ¢ = 3}, r= 6%, p = $500. 

2. If i= trp, find the value of « when ¢=4, r=6%, 
p= $50; when t= 27,4, r= 4%, p = $600. 

8. If d = discount, r= rate of discount, and p = price 
of goods, write the formula for d, as in Ex. 3 above. 

4. From the formula of Ex. 3, find the value of d if 
r = 331%, p = $270; also if r= 25%, p = $240. 

5. If a train goes m miles an hour, how many miles will 
it go in ¢ hours, at this rate? If d =the distance, write 
the formula. Find the value of d if m = 387}, ¢ = 33. 

6. If the population of this country increases r% every 
ten years, and is now p, how much will it increase in the 
next ten years? What will the population then be? 


14 USES OF ALGEBRA 


Roots. — The letters of,an equa- 
tion for which values are to be found are called ynknown 
quantities. \ These values are called the roots of the equation. 
To solve an ‘equation means to find its el 


15. Unknown quantities. 


For example, the unknown quantity in the equation + 3=7 
is x. The root of the equation is 4. 


16. How to represent known quantities. — The first letters 
of the alphabet, in an equation, represent numbers Supposed 
to be known. Cull gt 

For example, if told to solve the equation x + h= pre would 
take b from these equals, leaving 2 =a—b. Here the a and bd are 


supposed to stand for known numbers. 


17. The members of an equation. — The quantity to the left 
of the sign of equality is called the first member; that to 
the right, the second member. 


18. Symbol of deduction. — The word therefore is so often 
used in algebra that it has a special symbol (.’.). 


Since 20 =ao.e et aoe 


WRITTEN EXERCISE 


Solve the following equations : 


1: r+b=2a. 2.%—a=3b. 
Bay Sar a1 49. 4. 2—17=69. 
5. a+a= 125. 6. x — oc = 127. 
7, 19a” — 71 = 5. 8. 82 +92 = 190. 
9. 92 — 243 = 4238. 10. 17 x — 62 = 57. 
11. 142 +32 = 265. 12. 23a — 86 = 98. 
13. 261 +7 x = '303: 14. 9x + 126 = 621. 


. 426 +1382 = 501. 


Mal hile 25.2. 


bin 


EQUATIONS 15 


WRITTEN EXERCISE 


Find the value of the unknown quantity in Exs. 1-10: 


1. 7x = 609. 2. 122 = 204 ft. 
3. 132 = $221. fee a 7b. 
5. 1742 = 1218, GUIS 5 = 85.75, 
7. 123 y = 1353. 8.0 + 3 ft. = 7 ft. 
Bie 107 = 236, 10. $2 = 255 sq. ft. 


11. The average population per square mile in Africa is 
11 and the population is 126,654,000. How many square 
miles are there? (11a = how many?) 


12. Mt. McKinley is 20,464 ft. high, and it is 1606 ft. 
more than 3 times the height of Mt. Washington. How 
high is Mt. Washington? (3w +1606 = how many?) 


13. If r represents the mean (average) annual rainfall 
in inches at St. Paul, 7 + 33 represents it at New Orleans. 
If the mean annual rainfall at New Orleans is 60.5 in., 
what is it at St. Paul? 


14. The average balance of each savings-bank depositor 
in this country in a certain year was $417.21. This was 
$54.97 less than twice the balance of each depositor in 
Hungary. What was the average there? 


15. The number of English-speaking people in the world 
is 8.7 million more than twice the number of French- 
speaking people. There are 111.1 million English-speak- 
ing people. How many French-speaking people are there ? 


16. The average amount paid by each person in the 
United States for the general expenses of the government 
in a certain year was $5.96, which was $1.04 less than 
4 times the average amount each paid for our national 
pension fund. What was the average for pensions? 


16 TERMS USED 


SOME OF THE TERMS USED IN ALGEBRA 


19. Names of certain terms. —— We have seen some of the 
uses of algebra and have found that it is often helpful to 
represent numbers by letters. 

We now need to know the names of a few of the most 
important terms in algebra, especially those which we shall 
be using at once. 

20. Coefficient illustrated. ear the expression 2a, 2 is 
called the CURE AGTS ap 


Just as 2 apples means 2 times 1 apple, or 1 apple + 1 apple, 
sO 2x means 2 times 12, orlx+1z. 

That is, in 22, 2 shows how many times z is taken as an 
addend. 

Just as $2 means 2 of $1, 
So 24 means 2 of lz. 

The expression x means the same as 1 z. 

Liebe a 2 Xo = 10 an oe ean. 


ORAL EXERCISE 


‘Write the values of the letters on the blackboard before asking the 
questions. Give other examples of the same kind. 


1. If a=5 and b = 2, tell the value of the following: 
ab, 3a, 5b, Tab, a+b, a—b, 2a+b, 2a—bd. 
2. Ifa=2 and x = 4, tell the value of the following: 
ax, 5a,10%, 8ax,a+4,x—a,Ta+a2, 10a— dz. 
3. Ifa=2, b=3, and c=4, tell the value of the following: 
abc, a+b+c¢, ab, a+b, cb, c—b, e+a, 4b6—3e. 
4. If p=3,¢=7,r=9, and s = 12, tell the value of the 
following: 
ptEOndtr—-ss—r7r—3p,q—2p,S+p,p+r. 


COEFFICIENTS di 


ORAL EXERCISE 


1. If I have $x and you have $x, how many dollars have 
we together? 

2. If I have 2 dollars and you have twice as much, 
how much have you? 

3. What is the coefficient of x in each of the following? 
3x, 17 x, x (what coefficient is understood ?), 2a, 4a. 


4. If there are 8 classes in this school, and 0 pupils in 
each class, how many pupils are there in all? 


21. Coefficient defined. — A numerical factor written before 
a letter is called the coefficient of that letter. 

22. Letters may be coefficients. — We sometimes speak of 
letters as coefficients. If there are a pupils in this class, 
-and each one has @ dollars, they all have aw dollars. Here 
ais called the coefficient of x. 


Instead of one letter we may have several letters. Thus, in 
2 axy, 2 is the coefficient of ary, and 2 a is the coefficient of zy. 


WRITTEN EXERCISE 


1. If one bag of flour weighs x pounds, how much do a 
bags weigh? (Find the value for a = 25, x = 96.) 

2. If a glass jar of milk weighs y pounds, and there are 
xz of them in a basket, how much will the jars in a such 
baskets weigh? (Find the value for a= 6, #=12, y=1}.) 

3. If one chair costs a dealer c¢ dollars, and there are 6 
such chairs in a set, and the dealer buys a sets, how much 
do they all cost? (Find the value for a = 3, 6=6, ¢= 4.) 

4. Copy the following, writing beneath each the numer- 


ical coefficient : 
Zax, ¢ayz, 40% 2, 0.5ab, 652mn, 12.5 abcd. 


18 TERMS USED 


23. Factors. — The quantities which, when multiplied 
together, form a product are called the factors of the 


ert thi 


product. 
24. Squares. — If two factors are equal, their product is 


called the sguare of either. 


The product 2 x 2 may be written 22, called the square of 2. 
In the same way the product 25 x 25 = 25°, or 625, is called the 
square of 25. 


20. Powers. —The product arising from taking a quantity 
a certain number of times as a factor is called a power. 


— 


For example, the products 
Dire hae OTe, 3% oO KOKO seo Oreo k 

are powers of 2 and 3. 

That is, 25 = 32, the fifth power of 2, 

34 = 81, the fourth power of 35, 

6 = aaaaaa, the sixth power of a, 
48 = 64, the third power, or cube, of 4, 
6? = 36, the second power, or square, of 6, 


8 


26. Exponent. —In the expression «a°, 6 is called the 
exponent of ay and indicates the power to which a is raised. 


Tnid’, aina = 2, 0° G4 all =e and 0s) ae eee 


ORAL EXERCISE 
1. If a = 3, b = 2, tell the value of the following : 
A mM PCR raat tact “hae rhe lak) Be 
2. [fia = 2, b= 8, oe yb tell etheavaluemoL the 
following : 
a, 0b 860%, ba, a) RO eet, EE) 
3. In a’ and 2a, name the coefficient of a; the exponent 
of a. ‘Tell what each indicates. ys 


Abundant rapid oral drill, as in such examples, should be given. 


ALGEBRAIC EXPRESSIONS 19 


27. Algebraic expression. — A collection of letters, or of 
letters and other number symbols, connected by any of the 
signs of operation (+, —, X, +, ete.) is called an algebraic 
eXPVeESSION. 


For example, 3 a (since the absence of a sign between 3 and a 
indicates multiplication), 5+ 62, 2a?+ 367+ c¢. 


28. Term or monomial. — An algebraic expression con- 
taining neither the + nor the — sign of operation is called 
a eters 
a term or monomial. Gay 


orcas en RAED AP ABLE LD LLELLES AED : 


For example, the terms of 22? — 3a2y are 222 and 3ay. The 
expression 4 abz? is a monomial. 


29. Polynomial. — An algebraic expression composed of 
several terms or numbers connected by the sign + or — is 
called a polynomial. 

30. Binomial and trinomial.— A polynomial of two terms is 
called a binomial; one of three terms is called a trinomial. 


For example, a + 0 is a binomial, x — 3 y? + 22 is a trinomial. 


WRITTEN EXERCISE 


1. lia =2; 6 =5, c=1, find the value of each binomial 
in the following lst: a+ 07, ab-+ be, b> —40?, a+b. 

2. With the values given in Ex. 1, find the value of 
each trinomial in the following list: 3a? + 4 6c? + 3c’, 
§a?+267—-100c,a+b—c¢, 6a—b+4e. 

3. Ifx=7, y=5, 2 =2, write a monomial, containing 
one or more of these letters, that shall have the value 70; 
PoeeG0s 100; 25. 

i 4. If m=2,n = 3,¢ =7T, write a polynomial, containing 
one or more of these letters, that shall have the value 12; 
15; 30; 100. (For example, 5m +n" + 2 = 15.) 


20 TERMS USED 


SoME Usrs or MONOMIALS 


ORAL EXERCISE 


1. If a@=9, 6=8, what is the value of 4 ab? 

2. If a=11, 6 = 30, what is the value of ab? 

3. What is the area of a rectangle 3 in. by 4 in.? a in. 
by 6in.?. If area = a, find the area when a = 10,b= 30; 
Gea lO h == 20. 

4. What is the area of a rectangle @ high and 6 long? 
How does the area of a triangle of base 6 and height a 
compare with this? What is its area? 

5. If the area of a triangle is 4.0, find the area when 
a=10,6=20. If the measures are in inches, the area is. 
in what kind of units? 


6. The volume of a box 3 in. long, 2 in. wide, and 4 in. 
high is how many cubic inches? What is the volume of 
a box Z long, w wide, and # high? of a box a" by 6" by ce’? 

7. If volume = lwh, find the volume when 7? = 10, 
w=4, h=3. If tl, w, hostand for feet, hows willatic 
volume be expressed ? 

8. If the area of a circle equals m7?, where 7 = 35} 
and 7 stands for the number of units in the radius, find 
the area when r= 1. 


9. If the circumference of a circle equals 2 mr, find 
the circumference when r = 1. 


10. If a man saves $d a month, how much will he save 
in ¢ months? How much will this be if d=15 and t= 6? 
if d=.25 andt =10? 

11. If a boy earns ¢ cents a day, how much will he earn 


ind days? If he spends a cents in this time, how much 
will he then have? Suppose ¢ = 20, d=10,a=75? 


USES OF POLYNOMIALS 21 


SoME Usgs oF POLYNOMIALS 


ORAL EXERCISE 
. If we let a stand for 10, how may we represent 20? 24? 
If « = 3, how shall we represent 3 x 10? 30? 34? 
. If «=7, how shall we represent 33? 70? 74? 80? 


. If x=2, y=3, e=1, how may we represent 200? 


Pw nwo 


5. This room is a long, 6 wide, and ¢ high. What is the 
total length of all its 12 edges ? 

6. Suppose your marks in arithmetic were x on Monday, 
y on Tuesday, and 2 on Wednesday, what is the average? 
Suppose @ = 9,7 — 10, = 87 


WRITTEN EXERCISE 

1. Ifx=3, y = 39, what is the value of 372+ 5 y? 

2. A man earns a cents an hour, and 6 cents an hour- 
working overtime. What does he earn in an 8-hour day, 
working also 1 hour overtime? 

3. Rob earns 7 cents an hour, Jack j cents, and Tom ¢ 
cents. How much will all three earn if Rob works 5 hr., 
Jack 3 hr., and Tom 7 hr.? 

4. A room is Z ft. long and w ft. wide. In the middle 
is a rug r ft. square. How many square feet of the floor 
are not covered by the rug? 

5. Texas has 32,290 sq. mi. more territory than five 
times the territory of Michigan. If Michigan has m sq. 
mi., what is the area of Texas? Suppose m = 58,915? 

6. Richmond is 108 mi. north of the halfway point 
between New York and Savannah. If the distance from 
New York to Savannah is d, wae is the distance te 
Richmond? Suppose d= 904 mi.? 


THE NEGATIVE NUMBER 


THE NEGATIVE NUMBER 


31. Numbers below zero.— The mercury in a thermometer 
stands at 68° to-day. If it falls 23° to-night, we say 























a 
<= 
ra 
i} | pal 
fe 
y |= 
x 





that it stands at 45°. If to-morrow it should fall 
13° more, it would stand at 32°, the temperature at 
which water freezes. If it should then fall 32°, we 


=| would say that it stands at 0°, or at zero. When 
| it goes below zero, say 5°, we indicate this fact by 


using a minus sign, 5° below zero being written — 5°. 


32. Positive numbers. — The ordinary numbers with 
which we are familiar are called positive numbers. 


33. Negative numbers. — Numbers on the other side 


| of zero from positive numbers are called negative 


numbers. 


Negative numbers are written with a minus sign before 
them. Positive numbers need have no sign to distinguish 
them, but when it is desired to emphasize their quality a 
plus sign may be used, as in + 5, + 2, which means the 
same as 5, 2. 


34. Two uses for the plus and minus signs. — Hence 
the signs + and — have two uses: 

(1) To indicate addition and subtraction, signs of 
operation. 

(2) To indicate positive and negative numbers, 
signs of quality. | 

35. Debt and credit.— To say that a man has $100, 
or + $100, means that he has that amount to his 
credit; to say that he has $0 means that he is 


just even with the world; to say that he has — $100 is 
only another way of saying that he is $100 in debt, that 
he has $100 on the wrong side of zero. 


USES OF NEGATIVE NUMBERS 23 


ORAL EXERCISE 


1. If a man has $1000 and loses $900, how much has he? 
if he loses $100 more? $300 more? How is this written? 


2. If the mercury in a thermometer stands at 40°, and 
then falls 40°, where does it then stand? If it falls 10° 
more, where does it then stand? How is this written? 


3. If we call latitude north of the equator positive, how 
shall we designate south latitude? If we call latitude south 
of the equator positive, how shall we designate north 
latitude ? 


4, If a man weighing 170 lb. steps into the basket of a 
balloon pulling up just 170 lb., what is the combined weight 
of the two? Suppose the balloon pulls up 270 lb., what is 
the combined weight? | 


5. What is the altitude, in feet, of a point 1 mi. above the 
sea level? of a point 5000 feet lower? of a point 280 ft. 

lower still? of a point 100 ft. lower still? How shall we 
" indicate this altitude? 


6. If a man has $400 in the bank, and draws out $300, 
how much has he left? Suppose he then draws out $100? 
If allowed to draw $50 more (to “overdraw”), how shall 
we designate his balance ? 


7. If we call a certain point on the blackboard zero (0), 
and call distances to the right of it positive, how shall we 
designate distances to the left? If we call distances above 
it positive, how shall we designate distances below it? 


- 8. New Orleans is in 90° west longitude, which we will 
eall + 90°. What is the longitude of a place 80° east of it? 
of Greenwich, which is 90° east of New Orleans? of Paris, 
which is 100° east of New Orleans? How shall we indicate 
the longitude of Paris by using a negative number ? 


24 THE NEGATIVE NUMBER 


9. How much difference in price is there in selling a 
horse $15 below cost or $20 above cost? 


10. The temperature on one January morning in Denver 
was + 8°, and the next day it was — 4°. What was the 
average ? 


11. Weights of 7 lb. and 9 lb. hang over a pulley. 
ic] How will they move? What two methods could 


you use to make them balance ? 


12. A man who was $70 in debt paid $50. How much 
was he then in debt? Suppose he earns $50 more, how 
much is he then worth? 


13. If there is a house for every number, how many 
houses would you pass in going from 21 East Washing- 
ton Street to 5 West Washington Street, including both 
these houses? 


14. Jefferson Street is 6 blocks east of Adams Street, and 
Monroe Street is 12 blocks west of Adams Street. Monroe 
Street is how many blocks west of Jefferson Street? 


15. A game is played by throwing bean bags 
in the direction of the arrow. Suppose the score 
stands — 5, 3, 10,10, 0, — 10, 5,10,10, how much 
is the total score? 

16. The tide at the ocean is measured by a tide 
gauge. At mean (average) tide a pencil points 
to 0 on a scale, sliding to the right 1 space for 
every foot of rise, and to the left 1 space for every foot 
of fall. How many feet does the tide fall when the pencil 
moves from + 8 to — 3? 





t 


Teachers should give a great deal of oral drill of this kind, using 
blackboard illustrations when necessary, until the idea of negative 
number as the opposite of positive number is well understood. 


CURVE TRACING 25 


ORAL EXERCISE 


1. If we call a force pulling upwards + 3 lb., how shall 
we designate an equal force pulling downwards? 


2. Draw this figure on the board. Calling O zero, and 
the distances to the = : af 
right positive, point X’ O x 
to the distances +3; +5; —2; 0; —4;'+ 23; — 323 


1. 

36. Curve tracing. —In this figure is 
the successive days of the week a hated 
are represented on the line OX, / 
and the temperatures on the lines 
parallel to OY. The broken line 
shows that the temperature on one 
day, say at noon, was + 70°, the 
next day 60°, then 65°, 50°, 60°, 
Um O°. 





WRITTEN EXERCISE 


1. On ten successive January days in Duluth the ther- 
mometer at noon registered 40°, 60°, 55°, 50°, 20°, 0°, —10°, 
— 15°, 10°, 30°. Trace the broken line or curve. 


2. The increase in population in Nevada in four succes- 
sive census years was 36,000, 20,000, — 17,000, — 3000. 
Represent the time differences by 1 in., and 10,000 popu- 
lation by i in., and trace the curve. 


3. The increase in population in Virginia in eight suc- 
cessive census years was 200,000, 30,000, 200,000, 170,000, 
— 870,000, 180,000, 140,000, 200,000. Using half the 
scale of Ex. 2, trace the curve. 


Any tables of statistics or records of temperature furnish material 
for curve-tracing problems. 


26 ADDITION 


ADDITION 
ORAL EXERCISE 


. Add 2 apples + 3 apples; $2 + $3; 2a+4+ 3a. 

.- Add 4x+54;Txt+nx;244+9%; x+42; 24482. 
. Add 10x%+4+154; 25x+2; 54+22; 10%+4+ 0.252. 
. Add 2xe%+82+52;Ta+2a+a;3a+84a+8a+4+ 38a. 
. Add $8a+4a24 7x; 2n+3n+n; 4n4+2n+2n+38n; 
26+4b+6+6; 8m+m+m+im; 6c+4ce4106e. 


a fF ww DD & 


37. Like quantities. — Quantities like 2a, 4a, and 4a, 
that are the same except for coefficients, are called like 
quantities. 

38. Adding like quantities. We have found that like 
quantities are added by adding their coefficients, writing the 


letters in the sum. 2 he 

For example, 3 abr + 2 abr + abe = (8 +2 + lL) abx 2 abs 
= 6abzx, where the parentheses show that the num- 1 abr 
bers inclosed are taken together. 6 abs 


The sums of the following should be studied : 

2a $2 2 2abm 2 sheep 2s 
4a 4 4 4 abm 4 sheep 4s 
6a 6 g 6 abm 6 sheep 6s 
ieee ey 1,2 =4 12abm 12sheep 12s 








WRITTEN EXERCISE 


1. 7Ta+6a+23a+ 41a; 19% + 272 + 692. 

2.1744 3822%+u4+4+128e; 15a+49a+77a. 

3. 1506 + 16ab + ab + 28.ad; 182? + 97a? + 89 a. 

4. If a= 2, b =3, find the value of 16a+4 326. (Find 
the value of each separately, and then add.) 


ADDITION 27 


ORAL EXERCISE 


Add the quantities in Kus. 1-9, reading the columns 
rapidly, as you read a word: 


1. (ft. 3in. 2. 7sq.ft.4+3sq.in. 3. 7 f+37% 
Zits 410, 2 sq. ft.+4 sq. in. 2f+40 

Meocalss gi. pt. 5. bg--2g+41p 6.562.274 2 
9 gal. 1 qt. 9g+1¢q 9a+ y¥ 

7 4a4+264+3c 8 844+ y 9. 2w+sua+4+4y 
2a+36+5c OE ALD 2Zwtsax+tAy 
Ga 40 we 2y+42? 2wt38a+4y 


10. In adding algebraic quantities, as in Ex, 8, how do 
you arrange the like terms? Then how do you proceed ? 


WRITTEN EXERCISE 


Add in Hrs. 1-6: 


JL AB 2,.177%+31y 3. 63047 +19 y 
19a+.. 6b 92x%+ 73y a+ Ty 
36a + 396 4ja+ 9y 91a? + 80y 

4,.27@a¢+30+ c 5.16a+ 5b+7ece 6 a+ 64+ e 
4a+9d+4+2c og +9e s6a+386+3¢ 
9a+8b4+7T76e 5a+100 ba+56+5¢ 


Add in Exs. 7-9; then find the value of each addend 
and of the sum, lettinga = 2,b=3,c=5,x=1: 


eee 2b- Cc 8) a+ bee See Megs ra aR AG 
2a+36+2¢ 20 +2 20 Oa at 
a + ¢ db+a b6+2c+32 


b+ e¢ Za+4b s6+2c+ 24 


28 ADDITION 


EQUATIONS INVOLVING ADDITION OF ALGEBRAIC 
TERMS 


ORAL EXERCISE 


1. If twice a certain number plus 3 times that num- 
ber equals 50, how many times the number is 50? 

2. If 24+ 382 = 50, how many times x is 50? Then 
what is the value of x? 


In the following examples, first find how many times a 
certain number you have; then find the number, as above: 


3. 2+22=6. 4.¢+32=12. 

5.2 +92. = 20. 6. 2+ 8a = 81. 

Y bate (Grek that 8. 2 +62 = 63. 

9. «+192 = 80. 10. 2x2+ 62 = 56. 
lien 1 7 90. 12. 2y+13y = 45. 
13. 212 +42 = 78, 14. 388k —3k=70. 
15, Dia — 2 — 100; 16. 5% + 15m = 20. 
17, 1445-22 = 62: 18. ©+22+32 = 30. 
19. 27 2 — 7 2 = 100. 20..2+92 —22=—80. 


21. Twice a certain number plus 5 times that number 
equals 49. What is the number? 

22. A certain number plus twice the number plus 3 
times the number equals 60. What is the number? 

23. A certain number plus 7 times the number plus 12 
times the number equals 100. What is the number ? 


39. Illustrative problem.— Find the value of « when 
W7x+12¢%¢+2=510. 

1. Since 177 +1227+2=510, 

2. Then 30 x = 510, by adding like terms, 

3. And xz = 17, by dividing these equals by 30. 

Check. 17 X17 +12 x 17417 = 289 + 204 +17 = 510. 


EQUATIONS 29 


WRITTEN EXERCISE 


Find the value of x in Exs. 1-14: 

tT. 1924+ 27274+2= 94. 9 3oa2—2xr+7 = 168. 

a2 +2e2+3a2 = 102. sie Oa — 2. 126: 

Sow — 20 4+3.=12558 6. & 2a + Te = 30008 
.3+424+2xe=669. 8. 104+9x—22e=619. 
S60 +7 —25 = 275. 1050-92132 207. 

Il. ec+82+52=108. 12. 9x—6+e2+4+1=7)5. 

13. 2 --9xr+82=126. 14. 882+22+72+7 = 283. 
15. On account of change of temperature a watch gains 


27 sec. each day and loses 19 sec. each night. If set right 
to-day, in how many days will it be a minute fast? 


OO FF GO &w 


16. If in a class of 13 boys and 15 girls each pupil has 
the same number of cents, and the total amount is $2.52, 
how much has each? (1382 + 15 = what number?) 


17. A famous mathematician was once asked the time, 
and is said to have replied, “There remains of the day 
twice the number of hours already passed.” What time 
was it? | 

18. The length of a field is 11 times its width. The 
distance around the field is 36 rods. Required the 
dimensions. 

Tf x is the width, what is the length? The distance around 
is how many times the length and width? In cases of this kind 
always draw a diagram when reading the example. 


19. If you tell me to think of a certain number, then to 
multiply it by 3 (=3«), then to add to this twice the 
number (= 3a + 22), then to add 1 (= 3x+2x+41), and 
I tell you the sum is 51 (that is, 347+ 2x2+1= 51), how 
do you find the number? What is the number? 


30 ADDITION 


20. The Milwaukees won 83 games of baseball in one 
year, which was 3 less than twice the number they lost. 
How many did they lose? 

21. One of the best baseball records is that of Wagner, 
who was 512 times at the bat in one year. This number 
is 27° more than 5 times the number of runs he made. 
How many runs did he make? 


22. A man bought a suit of clothes and a hat for $36. 
The clothes cost 8 times as much as the hat. How much 
did he pay for each? 

If x equals the number of dollars paid for the hat, what repre- 
sents the amount paid for the clothes? for both? What is the 
equation? 

23. A man paid $1430 for a horse, a carriage, and an 
automobile. The carriage cost twice as much as the horse, 
and the automobile 4 times as much as the carriage. What 
was the cost of each ? 


24. A man paid $6600 for a village lot, for building a 
house, and for his furniture. The lot cost twice as much 
as the furniture, and the house cost as much as the lot 
and furniture together. What was the cost of each? 


» 


25. At Christmas Mr. Brownson gave to each of his 3 
children as many dollars as they were years old. Clara 
received twice as much as Helen, and Alden received as 
much as Clara and Helen together. All together they 
received $36. How much did each receive? 

26. I have a farm of 175 acres, of which 15 acres are 
used for buildings, garden, and woodland. Of the rest, 
3 times as much is devoted to oats as to wheat, twice as 
much to hay as to oats, and just as much to pasture as to 
hay. How many acres are devoted to wheat? to oats? to 
hay ? to pasture ? : 


ADDING NEGATIVE NUMBERS 31 
ORAL EXERCISE 


1. If a balloon pulling upwards 100 lb. is fastened to one 
pulling upwards 200 lb., what is the total upward pull? 


2. If a balloon weighing — 300 lb. is fastened to one 
weighing — 200 lb., what is the total weight? What does 
the negative mean ? 


3. If a man in debt $50 incurs a further debt of $20, 
how much is his debt? That is, if a man worth — $50 
adds — $20 to his capital, how much is he worth? 


4~ Add — $50. and — $20; — 70 and — 380; — 6 and 
—12. Make a problem about — 10 lb. added to — 30 lb. 
5. Add — 2a, —3a, — 6a; also add — 20 ay, — 30 xy, 
— 50 ay; also add — 6y, —y, —Ty, —38y, —Sy, —10 y. 


40. Adding negative quantities. — Therefore, to add several 
negative numbers, add as if they were positive, prefixing the 
negative sign to the sum. 


6. If a balloon pulling upwards 10 lb. is fastened to a 
12-lb. weight, what is the total weight ? 


7. If a balloon pulling upwards 15 lb. is fastened to a 
10-lb. weight, what is the total weight ? 


8. If in a tug of war, one class pulls 800 lb. to the right 
(which we will call + 800 lb.), and another class pulls 
750 lb. to the left, what is the resulting right-hand pull? 


9. If a man has a capital of — $500 and adds to it $700, 
how much is his capital? How much is — 500 + 700? 


41. Adding positive and negative quantities. — Therefore, 
to add a positive and a negative number, take the difference 
of their real values and prefix the sign of the numerically 


greater one, 


32 ADDITION 


ORAL EXERCISE 


Add the following : 
1. 6 ft. — 2 in. and 4 ft. — 3 in. 


2.6%—2yand4a—3y; alsol7z and «—y. 
16ab+2e 


3. 2 lb. — 4 oz. 4. 2x%—A4Ay a. 
3 lb. — 5 oz. on — OY 


—6ab+ 3c 


42. Adding several quantities. — In adding several quantities, 


some positive and others negative, it is at first 


convenient to add all the positives, and then the ie a8 oe 
negatives, then taking the difference of their real ihe ie 
values and prefixing the sign of the numerically a if es 
greater one. a 6 wes Ee 
Later, however, it will be found easier to read ae 
the columns, as in arithmetic. In this example oe e a 
we should think: “2’s, 6, 3, 2, 6, 9; y’s, — 5, —3, — oe oa ie 
—10, —13, — 11.” Practice soon enables one to teehee, 
read the columns much more rapidly than this. 
WRITTEN EXERCISE 
Add in Has. 1-26: 
1. o2ay + 32 2. 4m*n+ dmn? 
14 xy — 22 16 m?n — Tmn? 
—ry+ 2 — 9m’n — 18 mn? 
—16xy—5z 23 mn + 9mn? 
3. 142 ng + 148? 4. lba— 2+ ¢ 
19 pg + 37 ? 16 a? — 330? — 9¢? 
— 17 pq — 6738? (2 AU D2 -- 6c? 
5. 37a+ 926 —63c 6. 17abd+ 8be—4ca 
= O20. 1o0 pee 6ab — 5be+2ca 


25a — 19 b.-- 1S¢ —3ab+26c+2ca 


Li: 


ADDITION 33 


. 64arxy*z + p 8. m+ m—s 

78 @ay*z +4 p 3m*+4m 

81 axy?z + 9p Se 3s) 
fea OF 10. darwy+5z2+w 
a*—2Zab+ 6 ds acy +22—w 

4a*—4ab—20 Tary—32+w 

6pq+t+4qr+ os 129156 a 375 6/448 ¢ 

5pqg—4gqr+ 3s — 29384 —5966+17¢ 

9 pq +16s 489 a + 783 6 — 20¢ 


. 64 m?n + 92 mn? + 81, T6 mn? + 29 + 6 mn. 

. 28.07) + 382 ab?, 84 ab? + 29 a7b, 4207) + ab? 

. 82 pqr +45 grs + 4, 1296 grs + 487 — 39 pr. 

. 29 abed + 32 bede + 81 cdef, 64 bede + 92 cdef. 

. 81a—31b6 +48 0b, 696+52ab + 69a, — 385. 

. dla+42b—3c¢, 69a—38264+9c¢, —10b—6e. 

. 82p +424 — 349r — 89¢ + 47 p — 1277 + 8007. 
~27e@+14y4+72, —42e2—18y4+192, 182+ 62y. 
. 02m+17n—43p —18p + 3m —14n+4 92p +100m. 
. 8x2 — d4y4 722 —148y 4+ 79% — 144% + 832 — 2. 
. 8640 + 92774 3432, T83y—492 2, 629% —7T3y. 
. 892a—993b+444c¢— 82d, 1008a—7b+4 556c—18d. 
. 22a+6—8c+d—49e, 99b—d—e+3c4+ 78a. 
. 427 x? + 298 xy + 829y?, 827 a? + 43477, 298 x? + 87/7. 


Add in Ers. 27-29; then find the value of each addend 
and of the sum when a =1, b= 38, c =2: 


27. 


Z2a+4b— c¢ 98. 29 a7 +267 29° a7 bee 6 
3a+66—4e 10a? 3364 3b6+4¢ 
864+ 9e 4a*+ 76 Varese 


34 SUBTRACTION 


SUBTRACTION 


ORAL EXERCISE 
Subtract in Hxs. 1-5: 


1. $5 Zaft: BkiinG! 4. 5f 5. 5a 
$2 Pate 2d 2f Qa 


What must be added to the subtrahend in each of the 
following cases to make the minuend? 

6. 9x (pealiges 8. 19 abx 9. 238a+85 
8 8a 8 aba 38a+2b 


In Exs. 10-12 find the differences by seeing what must 
be added to the subtrahend to make the minuend: 

10. 23 sq. ft. 11. 23 f? 12. 18 42 +157? 

4 sq. ft. 4 f? Qx?+ Ty? 





13. In subtracting algebraic quantities how do we arrange 
the like terms? Then how do we proceed? 


WRITTEN EXERCISE 


Subtract in Hrs. 1-4, as suggested above: 


1. 142%+4 156y 2. 173 ax + 237 by 
9 ae+ b68y 96 ax + 142 by 

3. 648 2? + 2737/7 4, 12.46 abe + 2.83 
592 x? + 183 7 6.23 abe + 1.04 


Subtract in Hrs. 5-6; then find the value of minuend, 
subtrahend, and remainder, letting a = 2,b=1,c=8: 


5. 42a+ 610 6. 82a+1564+ 20c 
18a+19b 12a+15b+ 8c 


EQUATIONS 35° 


ORAL EXERCISE 


1. Ten times a number minus 3 times the number is 
how many times the number? 


2. If 10 times a number minus 8 times the number is 
42, what is the number? 

3. If 102 —2x”=16, what is the value of x? 

4. If 62+ 8x” — 2x” = 36, what is the value of a? 

5. Think of some number; multiply it by 6; to this 
product add 8 times the number; then subtract twice the 
number. Now tell me your result and I will tell you 
the number. How is it done? (Compare Ex. 4.) 

6. Make up a problem like the one in Ex. 5, using other 
numbers. Write the statement on the board, using x or n 
to represent the number thought. 

This presents an interesting mathematical game. Pupils may 


profitably propose such questions back and forth as part of the daily 
oral drill in solving equations. 


Find the value of x in each of the following: 


(Ei ie uentiy & 8 13a — 10a = Ss: 
9. 144 — Tx = 84. 10. 232% —182 = 65. 
lees Yel 0, 12; 522 — 212 = 938. 
13 oO ee OF. 14.9242 —81la¢=121. 


WRITTEN EXERCISE 


Find the value of x in each of the following : 

ee te 154. 9. 194-2 2 136: 

eee l4) 17 oe 526. 4. 14e—- 20 += 6b, 

beow +47 +1927— 223. 6. 241+ 82 + 167=—.289. 
7 13824+47274+6327=861. 8. 122+13874+1427=—429.- 


36 SUBTRACTION 


ORAL EXERCISE 


1. If a man is worth — $10, how much must be added 
to his capital to make him worth $0? $15? $50? 

2. If a man is worth — $100, how much must be added 
to his capital to make him worth — $50? — $75? — $25? 
+ $25? 

3. If the thermometer registers — 20°, how many 
degrees must the mercury rise to register —10°? 0°? 
30°? What is the difference in degrees between 10° 
—10° and + 15°? Show this by the help of the 
annexed diagram. 


20° 


4. Draw this picture on the board. Then show See 
what must be added to — 15 to equal 0; to equal5; ] 
to equal 15. 


5. Looking at the same picture, how much is the differ- 
ence between —10 and +5? That is, what must be 
added to —10 to equal +5? What is the difference 
between — 5 and + 20? 


6. State rapidly the following differences; that is, the 
quantity which added to the subtrahend makes the min- 
uend : 

$15 20° — 15° —102y 25 a*y 

— $10 aI BSS — 20° — 30 xy — 25 27 





7. What is the number which added to — 10 makes 0? 
makes — 20? makes —5? makes —15? (Use the above 
picture if necessary.) 


43. How to find the difference. —— Hence in algebra, as in 
arithmetic, 3 

The difference is found by finding the number which added 
to the subtrahend makes the minuend. 


SUBTRACTING NEGATIVE NUMBERS ot 


44. Illustrative problems. —1. From 327 —3y subtract 
22x" —2y. 


What quantity added to 222 makes 322? 824 — 3y 
Evidently 10z. What quantity added to — 2 y 224—2y 
makes —38y? Evidently — y. 10 gy 

2. From — 1027+ 7 y* subtract 62? — 5 7. 

What quantity added to 6 z? makes 0? Evi- —1022+ 7y 
dently — 622. How much more must be added 622— 57? 
to make —102?? Evidently — 1022. Hence — 16274 127? 


their sum is — 16 2?. 

What quantity added to — 5y? makes 0? Evidently 572. How 
much more must be added to make 7 y?? Hence their sum is 
how much? What is the entire difference? 


If pupils are not taught any artificial rules, but subtract in the 
way suggested, stopping at 0, as in the last example, if necessary for 
clearness, they will soon come to subtract negative quantities without 
difficulty. This is less confusing than to change signs and add. 


WRITTEN EXERCISE 
Subtract in Exs. 1-4: 


1. t4ay +17 yz+5 2. 15 ab — 20 be + ed 
Gay— 8yz—2 19 ab — 20 be — cd 


3: 17 m? + 1677+ 7 4. 62m?+15n?— jp? 
—17m?—167n?— 3 14 m? — 20 n? — 8p? 


Subtract in Exs. 5-8; then find the values of minuend, 
subtrahend, and difference, whena=38,b=1,c=4: 
5. 25a —176+ 3c 6. 35 a? + 406? 
32a— 9b—4¢ — 5a?— 300? 


7. 2744+ T56+ce¢+7 8. Fa-90' =33 4 
20a+70b—c— 3 = 20 SETESS 


38 





SUBTRACTION 





Subtract: 
9. Gabe +4 10, 38m? — 192" 
9abe — 7 4m? — 17 2? 
11. 42°-— 3y 12. 8p? + 4Apq 
9 x? — 27 y? 7 p? — 39 pq 
13. — 1243 + x* 14. a? — 147 xyz 
Be A ie lie x? — 298 xyz 
15. 23 a? + 91 a? 16. — 829 x? — 9 
49 a? — 37 a? 792 x? — 8 
17. 18 a7) + 4 al? 18. 439 a — 9276 
6 ab — 2 ab? 291 a — 2966 
19. 8m?+ Smnp 20. — 21.82%+0.9y 
6 m? — 11 mnp —42.9%—2 y 
91. =U60 + 0 ——c DO et ean ae 
—2Z1a—b6+6¢ 6a = 4a wa 
O30 ome eed 24. 42n*94+2¢4+7r 
6a? —5x2—1 —29p?—38¢q4+r 
25. —9x—Sy—9z 26. —92x+3y— 2 
—Tx—I9y—8z —T9x%—38y—9z 
27. From 8927 + 42 yz + 7 zw — 6 subtract 75 — 49 zw 
- 93 xy. 
28. From 822° — 4827+ 17 w — 37 subtract 342 — 93 x? 
+ AS 7? 2 3: 


Subtract, and then find the value of the minuend, sub- 
trahend, and remainder, whena=1,b=1,c=1: 

27a+92b—6c 30. 
—93a+486 —Te 


29. 


av+2ab+ 8? 


3a*—4ab—T7 0? 


REVIEW 39 


WRITTEN EXERCISE 


. Add v?+ 2ay+y*, v—2aey+y’, 827+ 877. 
. Add m?4+2m?—38m—1, m—2m?+m+1, m?+2m. 
Add 3abe—4bed + 5eda, — 6abe+ 5 bed — 5 eda. 
. From 3224+ 42?—5a”+6 subtract 42?— 16a” + 16. 
. From 2m?4+5n?—2x2+38y subtract n?-4xu+2y. 
seErom @ 26 — oe a suvtract a 40 + 6 ol ad. 
7. From 62?— Tay +44? subtract 5a? — Say + 3y?; 
then find the value of minuend, subtrahend, and remainder, 
tie and = 3. 
8. Add 3a7+4bc—6d, 4a*?—5bc4+T7d, 3a°+2 bc; 
then find the value of each addend and of the sum, ifa=1, 
Peas Coby. = 1 


Subtract, and then find the value of each minuend, 
subtrahend, and remainder, if a= 2, b=5,c = 83: 


G26 3070 2 a0 07 103590745 5c 
ola oe a0" Ge 0 Oe 


Add, and then find the value of each addend and sum, 
ufx=10,y=1,z=5: 


11. 84—57? +22 12. 5a? 4+ 2y7?—382? 
x—2y? a va + 32? 
9y?—4z e+ y+ 2 
13. 2a?4+ 4y?— 32? 14. xe*— Say+ 47? 
4o?— 6y?— 22? —427?+1bay— 37 
8a*7— B5y?+ 42? W7x?*— Bbry+52¥ 
ee — 132 1627-17 xy — 417 
—5a?+ 27? — 9x? — 137? 


16y?— 72 39 xy +17 4? 


40 PARENTHESES 


HOW TO USE PARENTHESES 


45. Use of parentheses. — The product of 2+ 3 by 7 is 
indicated by either 7 (2 + 3) or (2+ 3)7, the two having 
the same value, 35. The sign x is evidently unnecessary. 

If a=4, b=7, and z = 10, then (a + 6)z =11x 10 = 110. 


The difference between 7 and 2+ 4 is indicated by 
7 — (2+ 4), which equals 7 — 6, or 1. 

Ifia=9,7=5,andy= 2, thena—(@+y)=9—-—7T7=2. But 
a-z+ty—9—5 +2=4 72-6: 

The sum of 2+5 and 4+5 is evidently the same 
whether written (24 3) + (4+ 5)=549=14, or 243 
+4-+5=14; so the parentheses are not necessary. 

The quotient of 5 + 9 divided by 2 + 5 is evidently the 
(5 + 9) 5+9 
(245) or O45 But 

(64+9)+(2+4+5)=14+7, 
and this is not the same as 5+ 9+ 2-4 5, because 

46. It is the custom to perform multiplications and divi- 
sions before additions and subtractions unless the parentheses 
show otherwise. 


For example, 2+4 +2 means2+4=2 
but (2 + 4) +2 means6+2=8 

1 

1 








same whether written 


> 


2+4 x2 means2+8= 
but (2 + 4) x 2 means 6 xk 2 = 


WRITTEN EXERCISE. 
1. (xv + y) +m, where « = 10, Y= VOTO; 
2. (a+ 6)(c+d), wherea=1,b=2,c=3,d=4., 
3. 24+8~x 2, (2+ 8) 2, 2(2+4 8), 13 + 165 + 15. 
4. (8+7)+2,44+6+2, (446) +2, (7419) + (95 —82). 


REMOVING PARENTHESES 4] 


ORAL EXERCISE 


1. What must be added to 2a to make 5a? Then how 
much is 5a—2a? 

2. What must be added to 6 to make 0? to this to 
make a? Then what must be added to 6 to make a? 


3. What must be added to —é to make 0? to this to 
make a? Then how much is a — (— 0)? 


47. Subtracting negative quantities. — Let us consider the 
difference resulting from taking 6 —c from a. 
at+t0O+0 

We may writea + 0+ 0 for aif we wish. Then Rashes c 


what must be added to —c to make 0? to 8 to eee 
make 0? to 0 to make a? Then what is the value 
Ota — (0 — 0) 

Because a — (6 —c) =a —b + ¢, we see that 

48. If a quantity in parentheses is preceded by a negative 
_sign, the parentheses may be removed provided the signs of 
the terms within are changed. 

Because a + (6 —c) =a+b —e, we see that 

49. If a quantity in parentheses is preceded by a positive 
sign, the parentheses may be removed without change of sign. 

WRITTEN EXERCISE —G 2 f 

Remove the parentheses, changing signs where necessary : 
.@+2ab4+ 0? — (a —2ab +d’). 
m*? —2mn — 38 n? + (— m+ 2 mn + bn’). 
et Pade 9") = ag"): 
. @& — 8a) + (3 ab? — 0) — (8 +3 ab —3 ab? — 0°), 
a*b + be + ca — (ab — b?e + 0a) + (a 4-2 b?r — 3 ca), 
. 6a? + (3 y? —4 2?) — (a2 + 27) +. (yy? 42%) — (464+3 y 


42 MULTIPLICATION 


MULTIPLICATION 


ORAL EXERCISE 

», Multiply ybye2: 3b. 3462 soo ue een a 

. Multiply by 8: 30 mi.; 31 in.; 21 sq. mi.; 40 x; 50 aa. 
Multiply by.3: Yt: 2 in-3°7 squitezsqiin fia ay. 
Multiply by 7: a+26; 9a°?+50?; 8ab+c; 6abxe+d3by. 
Multiply by 6: 9 sq. ft. 2sq.in.; 9474 2y?; 2? + 67%. 
Multiply x* by 3; by 25; bya; by 5; by ab; by 2 abe. 
Multiply x+y by 3; by 17; by 200; by a; by 2a”. 


me) peek EA Lee OP heal oe 





50. Multiplying by monomials.— The multiplication of 
x+y by a is indicated by a (x+y), or by (w+ y)a. 
Hence a (x + y) =axz-+ ay, as in the following cases: 

Att Dani 400+ 2 dea+ 2y ot y 
5 5 5 a 
20 ft.10in. 2000+10 20%+10y ax+tay 





51. Therefore, to multiply a polynomial by a monomial, 
multiply each term separately and add the products. 


WRITTEN EXERCISE 
Multiply as indicated: 
, 23a by 125. 


. 43a by 27 y. 


u . A21 a by 23. 
3 

5. 16.ad by 15 2y. 

7 

9 


. 39x by 46 ay. 
. 16ax by 15 by, 
. 25 mn by 25 xy. 


ono - ww 


. 25 mx by 25 ny. 

. abe(a+y); 5(a+6). 10. ab(x?+y?); mn(a+b?). 
11. x(a+b); x(2a—6b). 12. b(p? +9"); 5a(m +2). 
13. pq (a+); b(@+y). 14. 8(2da.3hr.); 3(22+43y). 


LAW OF COEFFICIENTS AND EXPONENTS 48 


ORAL EXERCISE 


1. How many times is a taken asa factor in aa? in a?? 
mace ain a’? in.a*? in oes 

2. How many times is x taken as a factor inw?? in 2? 
iieceexere: (in 0708 Veiner ees 

3. Then what is the product of 070°? of 0904? of mm? 
Oe we OL Mm? OL mar Ber NOL meen 

4. If factors are the same quantity with the same or 
with different exponents, what do we do with the ex- 
ponents to obtain the exponent in the product? 


52. Law of coefficients and exponents. — Because 2x? x 3x? 
=2xx x 3xxx =62x°, we see that if factors are the same 
quantities with various coefficients and exponents, we 

Multiply the coefficients and add the exponents. 


53. The dot as a sign of multiplication.— In multiplying in 
algebra we not only indicate multiplication by the absence 
Ofeae sion bit also» by a, dot, thus: ax &= a-b'= ab, 
EXO ca os. 0: 

Multiply «? + 7? by xy. Here xy (x? + y?) = aya? + ayy? 
= 0°y +1xry*. 

WRITTEN EXERCISE 


Multiply as indicated in Exs. 1-10: 


ea 0-0, 2. ab-be-ca. 

3. xy(a + y). fare a O72 0. 

5. m?-m*®. m?. 6. 4a (x? + y*). 

7. 2m (m®+ 3 n’). Son tn 360 
9. am: 2am-3am. 10. 82a?(2a+ 3a). 


11. Multiply 22y(a@+y). Then let r=2, y=38, and 
find the value of each factor and of the product. 


44 MULTIPLICATION 


ORAL EXERCISE 


1. How much more shall I weigh with two 3-lb. weights 
in my hands? 

2. How much more shall I weigh with two — 3-lb. bal- 
loons in my hands ? 

3. How much more shall I weigh if I am relieved of 
the two 3-lb. weights ? 

4. How much more shall I weigh if I am relieved of 
the two — 3-lb. balloons? 


54. The law of signs in multiplication. — The addition of 
two 3-lb. weights adds 6 lb. That is, 2.3 1b. = 6 lb. 

The addition of two — 3-lb. balloons adds — 6 lb. That 
is, 2-— 3 lb. =— 6 lb. 

The subtraction of two 3-lb. weights adds —61b. That 
is, — 2-3 lb. =— 6]b. 

The subtraction of two — 8-lb balloons adds 6 lb. That 
is, —2-— 3 |b. = 6 lb. 

We therefore see that 

55. The product of two numbers with like signs ts positive; 
with unlike signs, negative. 


2%—3y 
Thus, to multiply 272 —3y by — 22, we have =p 
—224-—3y=62zy, and — 24-22 =— 42%. — 422 + 6 xy 


WRITTEN EXERCISE 


1. 3a(4a —2y). 2. — 4a(a? — 6). 

3. 2lay(a—12y). 4. 52a (a? + be). 

5. —15.a7b (a? + Dd). 6. — 41x? (a? — y?). 

7. 17am(—a +m), 8. loxy(— 3% — Ty), 

9. —a(—a—b—c—d—e). 10. — 21abe(— 36a — 425), 


11. —13m?(— 2m —38n). 12. wberd(— a? +b —c? +d). 


ao oa fF WH WD -& 


DIVISION 45 


DIVISION 


ORAL EXERCISE 


. Divide by 2: 4 ft.; $4; 4/; 4d; 400; 4h; 42. 

. Divide by 5: 25sq. ft.; 25 f?; 2527; 35 2?y; 45(x+y). 
: Divide by 7: 14; 2°739-(;2-7;7y; 63(@7+y' +2). 
SRI ICOM Ves Od Od aa ncaa eae Cred rey 
MUvIde Dy LYE OY, Oe CY OLY Fey ay ay, 1 Lou, 
» Divide 14 a’x by 14; by a7; by w; by 7; by 2; by 28. 


56. 


Law of coefficients and exponents. — Because 
Sie bl PAT gir oat 
je ee 
3x 3-4 





we see that in division involving the same quantities with 
various coefficients and exponents we 


Divide the coefficients and subtract the exponents. 


Oo mo Oo 


11. 


WRITTEN EXERCISE 


. TO p¢ + o pg. 2. 36 a7x77y? + 4a. 
32 a*b®ct + 2 ct. 4. 39 piq’ + 13 pig. 
27 a*x® + 3 ax? 6. 45 mn’ + 15 n°. 
28 men’ -- 14 n?. 8. 96 ax?z* + 4 azz", 
. 144 abed + 6 ae. 10. 1728 x®yoz? + 144 237328 


If 162*y* dollars are divided equally among «?z* 


people, how much has each? Suppose x=1, y= 2? 


12. 


If p=7+rt andi = 3007t, find the value of p. (Do 


you know what rule of interest is expressed by 7 = prt?) 


13 


. If c=187 (m being the Greek letter pz), and if 


c-+27=r7, find the value of r. (Have you learned what 
truth about the circle is expressed by ¢ = 2 7r?) 


46 DIVISION 


ORAL EXERCISE 


1. Divide 18 ft. 6 in. by 2; 18 f+ 67 by 2. 
2. Divide 28 lb. 8 oz. by 4; 28%+4+8y by 4. 
3. Divide 10 mi. 25 rd. by 5; 10m + 25r by 5. 


57. Dividing a polynomial ih a monomial. — Divide 16 a? 
+8a+6 by 2. 


Dividing each term separately, we 


my 2 
have 8a’ + 4a + 3. EAS ek ah 


8a@+4a+3 
Divide 35 a?y + 21 by 7a. 
7 2 
Dividing each term separately, we 12)35 xy + 21x 
have 5ay + 3. Sxy + 3 


Therefore, to divide a polynomial by a monomial, divide 
each term separately and add the quotients. 


a WRITTEN EXERCISE 
Divide: 


1. ax? + ay’ by a. 2. 25a? + 35 ¥ by 5. 
3. 8a + 245 by 4. 4. mnx? + mpy? by m. 

5. abe? + abd? by ab. 6. @+4a?+ 5a by a. 

7. 77 m+121 by 11. 8. 51 a? + 187 0? by 17. 

9. 2+ 223+ 32t by a 10. p’q?r? + 2 p?q?2* by p’q?. 
11. 327+ 396 ay by 3% = 12. 4a?y? + T6277? by 4x7, 
13. a§+38a2+4a?+ da dy a. 

14. 750° 4+ 25a?4 1254+ 625 by 25. 

15. aty3z? + 2 x8y2z + 3 xy?2® + 5 xy®e? by xyz. 

16. 125 x7y?z? + 600 xyz + 275 a®y%28 4+ 25 by 25. 

17. 21 m®> + 35 m1n + 56 mén? + 105 mn? by 7 m?. 

18. 33254 51 aty + 54257? + 69 x77? + 111 zy by 32. 


PARENTHESES 47 


58. The parentheses. — It should be remembered that the 
operations indicated within the parentheses are to be per- 
formed first. 


ee 
For example, x (x? + xy + y*) means that the sum of x? + xy + y? 
is to be multiplied by z. 
14 — (8 — 6) means that 8 — 6, or 2, is to be subtracted from 
14. That is, 14 — (8 —6)= 14-2 = 12. 


WRITTEN EXERCISE 


Perform the operations indicated: 
1. 2ax(a* + ax + x). 2. 4 mn? (mn? + 3). 
3. (1627y?+8xy)+4ay 4. (at*+a*)+a?+a(at+l1). 
5. (36 x8y?2 + 9 xy?z®) + 3 axyz. 
Grea) 07 0 6 a 
7. 32axy — 2axy+y(2ax + daz). 
8. (m> + m+ m) + m+m?(m?+1)+1. 
9. 25 mnz + dmnx — 38mnx + T mnx — 4mnz. 
10. 21 p°g + 42 p°¢.t+ p* (49 +39) — 2p’¢ — p’Q). 
11. (ax? + ax + 1) a*x? — (a*a?® + atx* + a®x®) + an. 
12. 6pg(p+q). Find the value when p = 2, ¢g = 3. 
13. 15 pq’ + 10 py? — 5 pg’ + 4 pq? — (6 py? — 3 pq’). 
14. 4a%b(a*°+ 0°). Find the value when a = 2, 6 = 1. 
15. x? (x? + y?+ 27). Find the value when «= 1, y = 2, 


16. (1628+ 82741224 20)+4. Find the value when 
e= ae : 
17. (a#+a*+a?+a)+a. Find the value of dividend, | 
divisor, and quotient when a = 2. 


48 | DIVISION 
ORAL EXERCISE 


.2u-8y; Gay+2x; 6ay+ dy. 
38a-—2b; —6ab+3a; —6ab+—205. 
—4a-5b; — 20ab+— 4a; — 20ab+ 50d. 


—3m-—4n; 12m7+— 35m; 12mn+—4n. 


ee hos ame 


59. Law of signs. — We see that 

Because + a-+6=+ ab, therefore + ab ++a=+0. 

Because + a-— 6 =— ab, therefore — ab ++a=— Bb. 

Because — a-+ 6 =— ad, therefore —ab+—a=+b. 

Because — a: — 6 =+ ab, therefore + ab +—a=-— J. 

That is, ; 

60. The quotient of two numbers with like signs is positive ; 
with unlike signs, negative. 


Thus, to divide 8222?—16ay by —162, — 1627)32 2° — 16 xy 
we have so Carte 
8242+—-16%=-—22, and —l6zy +-16z7=y. 


WRITTEN EXERCISE 
. 45 a7) +90. 2. 39 a°y? +— 13 ry. 
. 275 mn +— 25 n. 4. — 625 a7y?z? + — 125 xyz, 
. —1001L ay +Tay. 6. — 8535 p?q?r? + — 101 p*r*. 
. Axe — 32%)+-—2@. 8. (9 m2—18 mn 4+.27m)-+—9m. 
. (— 825 p? + 125 gq?) + — 25. 
10. (17 « — 512?4 1538 2”)+—172. 


11. Divide 32a*—8ab by —8a. Then find the value 
of dividend, divisor, and quotient if a =— 2, b=1. 


12. Divide 49 m3 — 63 m? — 84m by —7m. Then find 
the value of dividend, divisor, and quotient if m= — 1. 


onwynw Ww -_ 


REVIEW 49 


WRITTEN EXERCISE 


Add 45:22 + 19 2 2a endsed il (2 a? — 34 23. 

2. Add 78 p’q — 134 p77 + 7 p® — 139° and 432 pq’ 
+ 48 9? — 92 p’¢. 

Oo lrom ol + 1f a =346e¢-- 4072 © subtract — 42 24 
+ Sl 7 — 78: - 37 x: 

4. From 923 a + 431 «°y — 78 vy? + 127 subtract 34 
— 9244+ 78 xy? — 227 wy + 8 xy’. 

5. Multiply 2° + 422° — 3a* 4 8123 — 1252741732 
— 144 by 2a. Divide the product by 2. 

6. Multiply 2° —32*y4 4 ay? — 81 xy? + 26 yt —17 7° 
by —38ay. Divide the product by — 3ay. 

7. Divide 2m’ + 6 m®n — 18 m'n? — 292 m1n? + 38 m3n4 
—16m?n* by 2m*. Multiply the quotient by 2 m?. 

8. Divide 125 ay? + 625 xty* — 325 x8y? — 275 ay? 
+ 425 xy? by 25 x7y*?, Multiply the quotient by 25 xy?. 


Find the value of x in Exs. 9-21: 


O85 Lai ol. LO gal ae LY: 
ite) (Om 1.0 ab: 12.-” + 243 = 17. 
13 243 2. 14. 422 — 4 = 500. 
16°20 2S = 380. 16. 275 —2z2= 3. 
17, 34a +19 = 257, 18. 47 « — 58 = 600. 
19. “ ay 20. = oOo} raat a ma Ou 


22. If 110% of x is 84.70, what is the value of x? 

23. If 92% of x is $156.40, what is the value of x? 

24. What number increased by 10% of itself equals 143 ? 
25. What number decreased by 7% of itself equals 1953 ? 
26. What number decreased by 9% of itself equals 2912? 


00 FACTORS AND MULTIPLES 


FACTORS AND MULTIPLES 
ORAL EXERCISE 


1. What are the factors of 6? of 10? of 15? 
2. What are the factors of 2a? of 8x? of aw? of a‘? 
3. What are the factors of 9? of 37? of a?? of 352°? 


61. Factor. — The word factor is used in algebra as it is 
in arithmetic, the process of separating a quantity into its 
factors being called factoring. 

62. How to factor quantities. — We always factor a quan- 
tity by thinking of the quantities which must be multiplied 
together to make it. 


The factors of 21 are 3 and 7, because we remember that 
Sade at ONS: 

The factors of 3 a? are 3, a, and a, because we remember that 
DDO = do1G". 

The factors of abx + acx are a, z, and b+. 

Monomials are easily factored at sight. 


63. Factoring polynomials. — To factor a polynomial, ex- 
amine each term to find the greatest common factor. Then 
divide to find the other factor. 


For example, to factor 4 ay + 8 2?y? + 202y’, each term con- 
tains the factors 2, 2, x, and y. ; 

Hence, dividing by 4 zy, the factors 
are evidently 2, 2, z, y, and x? + 2 xy 
+5 y’. 


4 xy )4 xy + 8 x7y? + 20xy° 
eet ean y vt Oe ye 








WRITTEN EXERCISE 
Factor the following : 
Lu aa yoo ey =. 2. 14 m?nx®y; 91 m'n®z. 
3. ma+my;150°4+85a%y. 4. p2¢+pq?; 51 m>+17 mtx? 
5. 8 p®g +15 pp? +21 p77. 6. vy + vy? + v2? + 2? + a8. 


FACTORS 51: 


ORAL EXERCISE 


1. If a man is worth d dollars and loses rd dollars, how 
much is he worth? Factor the result. 

2. If a man is worth d dollars and gains 7d dollars, how 
much is he worth? Factor the result. 


3. If a man has p dollars and gains pr¢ dollars interest, 
how much has he? Factor the result. 


4. If you had w dollars in the bank a year ago, and have 
gained rv dollars, how much have you in the bank now ? 


64. Factoring certain formulas. — It is often more conven- 
ient to use formulas in factored form. For example, if 
some goods are marked m and the rate of discount is 7, 
the discount is rm and the selling price is m—~rm, or 

a 
m(1—r). That 1s, 







Veit m = marked price, and 
r = rate of discount, 
2. Then rm = the discount, and 
m — rm = the selling price, s. “} Agi. 
3. That is, s=m—rm / 
77) (1 = r). j 
4. If, now, m = $200, and r= 10%, we have 


s = $200(1 — .10) 
= $200 x .90 = $180. 


WRITTEN EXERCISE 


1. If 7, the interest, = prt, then the sum of the principal 
and interest equals p+ prt. Factor this. Find its value 
when p= $200, Dice PP ms 5%. 

2.) 1t's'—'selling price, ¢ = cost, and 7 ='rate’ of gain} 
write a formula for s. Factor it. Find the value of s 
when ¢ = $175, r = 20%. 


52 FACTORS AND MULTIPLES 


ORAL EXERCISE 


Name the greatest monomial factor in each polynomial 
Hy dU Rod PIG) 
1. ax + aby. 2. 2abe + 4 bry. 
3. 3pq7? + 6 qrs?. 4. 5m?n + 15 mn’. 
5. a®x + aby + az. 6. por + qrs + rst. 
. am +a'n+ ax + ay. 8. mx + nay + paz. 
. Sxyt+10xz4+2527?+50we. 10. Txyz + 2lwey + 3d5vwe. 


Oo a 


State the products of the quantities in Exs. 11-14: 


ll. «(@-+ Bd). 12. a(m* — n’). 
13. ab(a + 0). 14. m?n?(m? + n’). 
State the factors of the following: 
15. ax + bx + cx. 16. mx + my + mz. 
17. xy + ye + way. 18. 24°4+ 4ay + 6x2. 
19. abe + bed + cde. 20. 2 —3a?+4ax4 axyz. 


WRITTEN EXERCISE 


Factor the following: 


1, 2. m® — m?n. 

Soe 4. a®*b?x + aby. 

5. a? + ab + ab? 6. p®g?r — pq”. 

Vee ah ae 8. t+ 3épg +p? 

9. 82° + 427 + 2a. 10. 22+ 1527 + 16-2. 
11.3 mina 6 mina, 12. 27 ay + 9x7? + 38y. 
13. abc +- a°b%c + abc*. 14. 16 m? + 12 m2n + 8 m2. 


15. 5abed +10 cdef + 15c?d?. 16. 17 a? + 5labe + 153 a¥, 
17. 21 p79 +35 q’%p + 56pqr. 18. a*+322+ 4074 121 «x. 


MULTIPLES 53 


ORAL EXERCISE 


1. Name two multiples of 5; of 7; of a; of pq. 


2. Name two common multiples of 5 and 7; of 2 and 8; 
of a and 0b. 


3. Name the least common multiple of 7 and 9; of 6 
aneeee: Ol. 8 and 12 oredgandso: of ab and, bc. 


65. Multiple. — As in arithmetic, the product of two 
quantities 1s called a multiple of either. 


For example, ab is a multiple of a and of 6. 
66. Least common multiple.—The least multiple common 


to two or more quantities is called their least common 
multiple (1.c.m.). 


For example, abc is the least common multiple of ab and be. 
In algebra this is often called the lowest common multiple. 


67. Finding the least common multiple. — The l.c.m. is 
usually found by factoring. 


For example, to find the l.c.m. of az + ay and bz + by, we have: 
az-+ay=a(x+y), and br + by=b(z-+ y); therefore. the l].c.m. 
must contain the factors a, b, x + y, and is ab(x + y), or abx + aby. 


WRITTEN EXERCISE 


Find the l.c.m. of the following: 


1. abz, bey. Pde Coie HO 3. px, gary. 

4. pgr, q’s. 5. mnw*, npry. 6. ab?c, a*be?. 

7 at, ax + be. 8. yp? + pq, 9 + Gp. 

9. abx + aby, abe. 10. 27 a® — 27 0°, 9 ab. 
11. a(x — y), bx — by. 12. cx(m+n), y(m+n). 
13. 41 a? + 82 b?, 123 ad. 14. m®n + mn®, mp + np. 


15. am + abn, m* + bmn. 16. 2°43 2%y, 3 my + mx. 


o4 FACTORS AND MULTIPLES 


WRITTEN EXERCISE 


Factor in Exs. 1-22: 
. 39 abe + 65 abc*. 2. m>n*p + mn7p'. 

. 17 u’v — 119 we? 4. pgr — gqrs + rst. 

e+ 3xe*y+5 ay”. 6. 252 ay? + 84 ay. 
.a+5a% —4 ab, 8. 86 ay + 129 x7. 

. abed + bede + defg. 10. « — 32a*y 4+ 3a”. 

11. 27 amn + 108 mnp*q. 12. 57 m'n? + 95 m?n’. 

13. m?np + mn?p — mnp*. 14. m* + m?n + 4 mn”. 

15. pi—-3p?—Apt+ 5p. 16. 627+ 82y + 102°. 

17. — p’g?r — py’? — par. 18. 82 p®g?r — 123 p?q??*. 
19. 42a7b —65.ab?+168 abe. 20. 32 22 + 72 xy? — 128 x. 
21. 37 p’¢ +185 p7r—TA4pg. 22. —82°y2—627y22—4ayz 


Oo F oO WO — 


Find any multiple of the quantities in Exs. 25-31: 
23. pr. 24. 3¢q*re. PA Leder Pies 
26. a+. 27. “1 — y. 28. m? — n?. 
Pas ahi vinta bo mt 30.-07 Zab b*  3h. a — o0--< 


find any common multiple in Hxs. 32-37: 


32. ab, 3 be. 33. 2 pq, 3 qr. 34..°, n°, 2:7. 
35. a*, a? — 6. 36. m*, m + n. 3722070, 6 abt 
Find the lem. in Exs. 88-49: 

38. 2 a7b, 4 ab? 39. a* — b, 15 ab. 

40. a+b, 25 ab. 41. p+ 2p, 27 p*. 

42. ab, be, ed, de. 43. 15 pq?r, 20 nar. 

44. 32abe, 2a+6. 45. 2a, 36, 4e, 5d. 

46. abc, bed,a+b+e. 47. 15 p??r,ptatr. 


48. a? + 2ab + 0, 3 a? 49. 2a? — 6ab+ 26% 2a. 


FRACTIONS aY9) 


FRACTIONS 
ORAL EXERCISE 


1. What are the terms of the fraction 2? Which is 


3 
the numerator? Which is the denominator? Answer the 


same questions for the fraction A 


2. In the fraction 3, into how many equal parts has 
the unit been divided, and how many have been taken? 


Answer the same questions for the fraction = 


3. If we think of 18 as an expression of division, which 
number is the dividend? Which is the divisor? Answer 


the same questions for the fraction ; 


68. A fraction.—One or more of the equal parts of any 
unit is called a fraction. 

A fraction may also be considered as an expression of 
division. | 

Thus, 2 means 2 of the 3 equal parts of 1, or it means that 2 


has been divided into 3 equal parts. ; means a of the b equal 
parts of 1, or it means that a has been divided into b equal parts. 

69. Terms of a fraction.— The terms numerator and denom- 
inator are used as in arithmetic. 


70. Integer. — An algebraic quantity in which no fraction 
is expressed is said to be an integer, or to be integral. 

For example, az + 0 is an integer, and is integral. 

71. Sign of the fraction.— The sign written before the 
fraction is called the sign of the fraction. 


a 
a ; : bp 
For example, — 2 is a negative fraction, and 4p 5 positive 
fraction. 


56 FRACTIONS 


ORAL EXERCISE 
1. In reducing }8 to lowest terms what factor is canceled? 
What is the result? Answer these questions for ari 
2. Reduce each of these fractions to lowest terms: 


mx abu? A x*y 14 pgr 
my aby 6 xy 21 p*qs 








72. Reducing fractions to lowest terms. — Fractions are 
treated in algebra just as in arithmetic. Jractions are 
reduced to lowest terms by canceling all SORES common to 
numerator and denominator, 
me pe 


~_ 


18 reduced to Bihay 








Thus, the fraction : by canceling 


the only common factor, z. 


WRITTEN EXERCISE 


Reduce the following to lowest terms: 





2h a? 2 3 
hy ee get eee gue a 
b?a bx2y? wy? arx*y 
a abc? 5. eseedh # 16 a*x*y | 
a®b’ec 3d any? 36 alyz? 
aa? + ay? ary + ye par + girs 
1. — 8. ———_—_-- 9. ————"—__- 
Bs axy pars 
2G 0 51 m?n+ 85 mn? ote oO OU 
au 4a at 17 mn te axe) 
13 a? ag 8 a2h =k 3 ab? 1 pg + p97 + p97 +. py 
ab par 


mn + mn + mn? 81 vty +108 x74? + 207 x y* 


ao mn i 27 xy 


USES OF FRACTIONS OT 


WRITTEN EXERCISE 


1. If the perimeter of a square is 16, what is each side? 
What isthearea? If the perimeter is p, what is each side? 
What is then the area? 

2. If the perimeter of a rectangle 16 in. long is 50 in., 
what is the area? Suppose the perimeter is p and the 
length is 2? | 

3. If a piece of silk of a yards is worth 6 dollars, how 
much will an employee of the store have to pay for a yard, 
allowing him a discount of ¢ cents a yard? What is the 
resulvaita = 40.-bi=2:60,6 = 257 

4. A car wheel is 6ft. in circumference. How many 
revolutions does it make in going 60 ft.? in going d ft. ? 
If it is ¢ ft. in circumference, the number of revolutions, 7, 
in going d ft.,is how many? That is, what does 7 equal ? 


5. If a barrel weighing w lb. is rolled up an incline s ft. 
long, to a point d ft. high, a power of we lb. is exerted; 
that is, pa. How much power at f 
be used to roll a 200-lb. barrel up a 10-ft. ae 
incline to a height of 4 ft.? 


6. How much power must be used to roll a 100-1b. barrel 
up an 8-ft. incline to a height of 4 ft.? also a 150-lb. barrel 
up a 12-ft. incline to a height of 2 ft.? also a 300-lb. barrel 
up a 15-ft. incline to a height of 5 ft. ? 


7. Two cogwheels, one having 9 cogs and the other 27, 
are fitted together. How many times will 
the smaller wheel turn for each turn of the 
larger? How many times, if the larger has 
10 cogs and the smaller 5? if the larger 
has a and the smaller 5? 


08 FRACTIONS 


ORAL EXERCISE 


2 ; 
1. How do we reduce 3 to sixths ? . to dcths ? 


2. Reduce to fractions with denominator abzx: =) 5 = 
x a 
3. Reduce to fractions having the least common denom- 


= and 35 andi: 


; 2 4 
15 — 
inator: > and:= 53 3 5 5 


5) 


73. Reduction of fractions. — As in arithmetic, 

Both terms of a fraction may be multiplied, or both divided, 
by the same quantity without changing the value of the fraction. 

2 LO Ue ee cleus OC me See 

For example, -~=—, —=-, -=—, —=-- 

See Eee AI ee i ere] 

74. Least common denominator.— The least denominator 
common to several fractions is called their least common 
denominator (1.c.d.). 

This is also, in algebra, called the lowest common denominator. 

For example, the l.c.d. of oe ot and ms is bac?. 

pars: ce 

75. Finding the l.c.d.— As in arithmetic, the l.c.d. is 

evidently. the l.c.m. of the denominators. | 


a 
For example, to reduce ; 


the l.c.d. ahd 2b 

The l.c.d. must evidently contain the 
factors x + y, 2, b, 4 (which contains 2), 
and 6? (which contains b). It is there- 
fore 4 b? aoe a). peices both terms 55 Lie yy 
of by 4 8, we = by 2 2b(x + y), and m m(z+y) 

as 402 40? (rey) 
of = by x + y, we have the results. 





and ar to fractions having 
2 


Oye ae) 4.007 
aty 4P(+y) 
a 2ab(x+ y) 





REDUCTION 59 


WRITTEN EXERCISE 


Reduce to fractions having the denominator indicated in 
parentheses in Hxs. 1-10: 




















a p * 
e SS 4 8 e e 2 2 e 
1 3° § b°) 2 apr (L297) 
m ‘ory x 
3. ep (m?n®p*), 4. Bo (2 pq’). 
& 2 ap fe 2 
5. pare (d -t be). 6. oa Fay (2 (a 2 qr). 
a+b taal 2 Qayd pplad 
7. mas. be ence 8. Pate (x?y72? + yz), 
m 
9. Apna (ab?m? — ab?n’?), 
x 
1035. oe + Oe 6 wt). 


w— 2 we + 2 


Reduce to fractions having the Le.d.: 


























Gh ¢ Cee Tia 
es: ee ee 
we bc a 2b 2n bn 
1 1 1 
13. Ce ae ee 14. ie 2 2,2 
a a2 a? gq? 7? jemi 
fi : 
fishy 2 eee i 
oy o Tr ee kee ts ee 
3 ah ee 
7. ———, ieee Ae 
Wet | 1 pqr. ars. rst 
GoatD (ae 0 (ieee Dad gn oe 
19, ee 72 oe hah, cA a+b b Bb 
ak BLT & a i 22. af ) we ’ ie . 
a—b+e b e¢ dspqr Oagrs Ipqs 


AS 
ae 


ip 6 


Ui eee Vee eae 
CREST. Fu 2ab 4c 


Ors 


60 FRACTIONS 


as 


ORAL EXERCISE | 
1. How many fifths in 1? in 12? in 2? in 24? 
2. How many Uths in 1? in 3? inm? inm+ ae 
3. Reduce a to aths; a + ; to yths; 2° + . to y*ths. 
76, Mixed quantities. — As in arithmetic we speak of 23 
a mixed number, so in algebra a +5 is a mixed quantity. 


77. Reduction to fractional forms. — Mixed quantities in 


algebra are reduced in the same way as in arithmetic. 


1 


2 


* 25. 


Arithmetic: Algebra: 
Reduce 3¢ to fifths. Reduce a2 + to dths. 


Since 1 = 2, therefore3 =15. .1. Sincel = us therefore x= me 





seg Goes sae, Te 1s 


S a>va d 


WRITTEN EXERCISE 


Reduce to fractional forms: 


1 


3. 


5. 


it; 


2 

+7 to gths. 2. 3m? + = to kths. 

2a% +0+5 to bths. 4. a? —3ab + 0 to dths. 
m+ 3mn+2n* to nths. 6. + Bary +9? +7 to yths. 

7 mn 

Spi 4. .4mn+—-: 

p +39 8 4 min + a 
.4a4+2d4-. 10. 16.2? + 15.0y +2 


BD garetts ante 12. 37 a? — 1508 4 464. 
25 y 5a 


IMPROPER FRACTIONS 61 


ORAL EXERCISE 


Zz 2 
1. Reduce to whole numbers: 5; ue Bat. om 
2a 8 a 5 
3 
2. Reduce to whole numbers: OL SO y SEOs nO 


a 3m? 
3. Reduce to mixed numbers: 5. iF Cts 1, x3 
ye ns a a 


78. Remainders in division. — In algebra, as in arithmetic, 
a remainder in division leads to a fraction in the quotient. \ 





Arithmetic: Algebra: 
Divide 124 by 5. Divide «7+ 38x41 by a. 
5) 124. LY eer aby 
24, 4 remainder, x +3, 1 remainder, 
or 244. or xz (oe eee 
ex 


79. Reduction of improper fractions. — Zo reduce an im- 
proper fraction to a whole or a mixed expression, divide the 
numerator by the denominator. 

For example, 2) +2a?+3¢+4+1 

Spo We] } 1 
Geteodet St tel es ho waa, w+2e4 +38 a 
ZL 


x 


WRITTEN EXERCISE 


Reduce to whole or mixed quantities: 


L 20+ 52° +1 n Pages 
% Pp 
16m? + 32m+7_ 4 4dat+6a°+3 
16 m DW fe 
Sa + 407+ 20+ 6. 6 12 a? + 60% +176 


22 Ga? 


62 


FRACTIONS 


WRITTEN EXERCISE 


Reduce to fractions having the denominators indicated: 


2m aa 


2: 2 
AC aoe) 3. ae (15). 


ae, (81). 5. se axa 6. imey rey 








b) (18 LYZ), 


eee 





é ee 
| atte, (49). 8, wat, (25a). 9. 


» (ayz). 


Reduce to fractions in their lowest terms in Exs. 10-18: 























A ng 21 a*b Lianne 
Ne 16 pq’ te 49 be a 51 mn? 
125 abc? 2p? +3p 2 x 
Loboonathe eng pt2p Lae eens 
2s " 3s0x%+ 35y 38m? 
$c eo Sear th 49 ot 9m? + 27 m 
Reduce to whole or mixed quantities in Kxs, 19-27: 
2,,2 or 87,2 AS) 54 
19, ey’. 225 alb™ 91, O28 RT. 
9 xy 25 ab 125 p*q?* 
4 b 2 3 = 
5 5a ath a 81 a*y +19 04 32 +5. 
9a 9 xy 32x 
24 x? — 2: Ole e 5 PAT 
9 24% ier 32 Py? + 15° 27, (x? +81 0+25 | 
6% 8 xy 9 x 


Write the Sea in fractional form: 


28. iy anes 








= 


30. oboe Bt 
d y 


32. 21 ot 17at + 33. 82 xy + 21 xy? + —. 
x y 


FRACTIONAL EQUATIONS 63 


ORAL EXERCISE 


1. If half of your weight is 40 1b., how much do you 
weigh? If 4 of x is 40, how much is x? 


2. Find the value of x in each of the following statements : 


is ay 5 ec 
ed 5 = 15 3 = 50 3 = 10 Pier at 
: re 5 +1=11, what does 5 5 equal? What does x equal? 


2 
If = Bei i = 4, what does 2 equal? What does a equal? 


4. If you have equal weights in the two pans of the 
scales, will they balance if you add 2 oz. to.each? subtract 
2 oz. from each? multiply each by 2? divide each by 2? 
State four operations which you may perform on the two 
members of an equation without destroying the equality. 


80. Multiplying equals by equals to avoid fractions. — In 
a= 1, if we ee these equals by 3 we 


the equation = 
vw 


ae e@= 21. Inthe equation * <= +6=8we have ~- oie = 2, 
ors =1; ; multiplying by 5, x = oe 


5 
81. Clearing an equation of fractions. — Multiplying both 


members by such a number as to make fractional terms 
integral is called clearing the equation of fractions. 


WRITTEN EXERCISE 
Find the value of x: 


1 Z+2=9. 2. = +2= 30. 3. 45 =15, 
32x oe 9a a 
4. 7 t4= 1s. et, 2 6. aq 8 19. 

20 5a 


64 FRACTIONS 


J am thinking of a number whose half added to 71 
amounts to 97. What is the number? 


tL x = the number, 

2. Then > = half of the number. 

3. Then = + “1 = 97, by the statement. 

4. Then : = 26, by taking 71 from these two equals, and 
5. : = 52, by multiplying these equals by 2. 


Check. If I add 71 to 26 (half of 52), the sum is 97. 


82. Checking the result.— Always check by placing the 
result in the original statement. You may have made a mis- 
take in getting your equation as well as in solving. 


WRITTEN EXERCISE 


1. What is that number 2 of which, plus 5, is 45? 

2. What is that number 4 of which, less 6, is 10? 

3. If I add 10 to 5% of a certain number, the result is 
30. What is the number? 

4. I'am thinking of a number such that its seventh less 
6is 27. What is the number? 

5. If I subtract 6 from 7% of a certain number, the 
result is 15. What is the number? 

6. If to acertain number I add half of the same number, 
and then add 7, the result is 10. What is the number? 

7. If from a certain number I take 4 of the same number, 
and then add 10, the result is 30. What is the number? 

8. Make up a problem somewhat like the above, and 
then solve it. 


Teachers will find that the pupils will derive much benefit from 
making original problems as here suggested. 


ADDITION OF FRACTIONS 65 


ADDITION OF FRACTIONS 


ORAL EXERCISE 


Dan wi 32. cee b 

il Add ~ and ~; 7 and = ; and 
BeAdd ahd <: - “vende eee and 

x wx x x 


83. Addition of fractions. — Fractions are added in algebra 
in the same way as in arithmetic, by first reducing to the 
least common denominator. 


























Arithmetic: Algebra: 
3 1 a c 
Add — and =: Add — and —: 
tices G pen Gol 
1. The least common denom- 1. The least common denom- 
inator is evidently 12. inator is evidently bed. 
2. a 9. eae 
uh iy. be bed 
1 2 2 
and == =~. and ee 
6 11 bd bed 
it 1+ 2 
oO. Lhe. sum, = ——- 3. The sum = AOS : 
12 bed 
WRITTEN EXERCISE 
a6 mn 
ne oe ai 8 
abe abe e abe b?ed 
3. — . reo one =: 
yz way 2py 8 py 
5 pimp 4 ab 6 C04 4b? 
‘4m%a 3 mn? " 4 a¢ a? 
b ee ty 5 ebay? = 17 82/? 
ij SEE 0, aati pees fee 
Cc 2.0 6 mn 36 mn 


66 SUBTRACTION OF FRACTIONS 


SUBTRACTION OF FRACTIONS 


ORAL EXERCISE 








Tie at ome eed Ae EER GR Fe 
p¢4yl 18 8a a tbe 3 ae 
w24eh OLIVE 26. 6° 2a y°>m 3m 


84. Subtraction of fractions. — Fractions are subtracted in 
algebra in the same way as in arithmetic. 














Arithmetic: Algebra: 
3 1 a+b het | 
From — take —=- From take . 
4. 6 « 2¢ be 
1. The least common de- 1. The least common denomina- 
nominator is evidently 12. tor is evidently 2 be. 
* 3. -9 Git0 7 Gab eb? 
7 me Vee ov Te eRe. 
1 2 a—b 2a-—-2b 
and i, d SS 
ee ee Tb) ee he Dihe 
7 wb + b2—-2a42b 
3. The difference = —. 3. The diff. =~ — : Bee 
2 2 be 


85. It should be noticed that the fraction bar has the 
same effect as parentheses, the 2a—2b in the above 
example being subtracted as explained on page 37. 


WRITTEN EXERCISE 














eee e TE oye 
ar jee 

3 2S tS 4 Sab dab 

Sikes 107, 6 070 seca? 

5 a@_a—db Re ered 
wi) b '3mn? 2 mn 


21. 


23. 


25. 


27. 


ADDITION AND 


SUBTRACTION 


WRITTEN EXERCISE 


Add and subtract as indicated: 














xr 2 

ah ab? 

ey gale 

Ge 0d —— 0 

Bab?) 3a 

a b a—b 

a? pec ee 

im aie sale 
abe aa 


Hig ON a LV 
a0 Dg log 


iran Yomneee = 1s 











abe bed 
tO yee hOncke Cctak th 
abe bed 
Clans 21 soot ard Siglo 
x y Zz 
a+2ab+b?  a?—2ab+l? 
ab ab 
eee (eee eee 
ieee rad ee 
y Zz x 


pee a ee se 
PY qr 


UVW VUWX WLY LYe 


2,2 


Goi yz 


abe Migd SEe ee 





























ripe ie tap 
py mg 
fi mn mn? 
TE Fae 
bx x 
oer ar 
mtn m—n 
3s —— : 
im 30 nN 
2 2 Bs 2 
10. Porgy Ga! 
Pq’ qr 
Ce be eae 
Le En pie Se 
" a + ¥? a? — ¥? 
Ve AD Ay 
Cae Ore Co ade ne 
Ean ee eae 
18. Car AM wie 
ab be Cu 
Qo (i= Lee a ey 
20. erie ioe a ie 
29. Le Wg ee 
b a 
24. Garnnal TU TMA 


C 02 ac 


Oo 0 ee a 


ab? Nabe c7a? 





(U4 LY TROND View an cea ciel Se ETE 


22w? 


68 MULTIPLICATION OF FRACTIONS 


MULTIPLICATION OF FRACTIONS 


ORAL EXERCISE 


1 er ae ee) Me 
1. How much is twice 3° a times —? m times —? 


2. How much is 5 times “eo 
15 y 


b 


(Cancel.) m times = ve 


3. How much is 10 times i (Cancel.) mn times ae ? 
15 my 


4. Tell how to multiply a fraction by an integer. 


86. Multiplication of unit fractions. — It appears from the 
above exercise that we multiply fractions in algebra in the 


same way as in arithmetic. 


For example, consider unit functions: 


Arithmetic: 


Lee 
Take 3 of 5 
We think of 1 as divided 
into 5 equal parts, and of each 
part as divided into 3 equal 
parts, so that 1 is divided into 
5 x 3, or 15, equal parts. 








Algebra: 


yy i 1 

Take b of rE 

We think of 1 as divided 
into d equal parts, and of each 
part as divided into 6 equal 
parts, so that 1 is divided into 
b- d, or bd, equal parts. 





Therefore x of M = rate Therefore A of L = sat 
OF SOLO bees bd 
WRITTEN EXERCISE 
1 —b 2 
loa: 2. — 2. —. 3. at 
hy LY e*d* 
4 a a 1a) a—b 
Le re n 16ay cd 


MULTIPLICATION OF FRACTIONS 69 


ORAL EXERCISE 


1 How much is 5 of 5? 5g ? = 


b 


a 
Co 
, of 5 


2. How do you multiply one fraction o another ? 


87. Multiplication of fractions. — Compare arithmetic and 


algebra again, as on page 68. 


Arithmetic: 


nen 
Take 3 of 5 


Because 4 has been divided 
into 5 equal parts, and each of 
these into 3, therefore 4 has 
been divided into 15 equal parts. 
4. 


1 
Therefore 3 of — 5 Soe 


Beek 
0 Oe 
Therefore = 0 5 


Algebra: 
a ,e€ 
Take 5 of 7 
Because c has been divided 
into d equal parts, and each of 
these into 6, therefore c has 
been divided into bd equal parts. 


Therefore — : of | = rake) 
b Phd 

here ore it —-=a: pep cae 
b ad bd bd 


88. Therefore in algebra, as in arithmetic, 


To multiply one fraction by another, multiply the numera- 
tors for a new numerator and the denominators for a new 


denominator. 


89. Cancellation should be employed whenever possible, as in 


ET es we 


“the case of. ts Hews 


a b 








WRITTEN EXERCISE 

















a c? —@ —€ Cau ec 

Us Be de 2 ak nm nn 
A abe abc? Sap 3 YS abe xyz 
" m?nx mn? Saye -2ab eye abe 


ae 


70 MULTIPLICATION OF FRACTIONS 


ORAL EXERCISE 


1. At m miles in # hours, how far will a train go in 
GRU ge phakere Sina" Abaya Yipiininign pRIVI/!1ehe 


2. At m miles in / hours, how far can you walk in 


; Of an Hour ino years: IN 2 nr s 


3. If y yards cost d dollars, what will 1 yd. cost? = yd.? 
m yd? 

4. A man earns d dollars a week for working 6 days, 
h hours a day. How much is this per day? per wo 
_ 22 


5. In a circle of circumference ce and diameter d, — a 7 


foUnoo rer ue ? 

d 

6. If a acres of land are worth d ee how much is 
1 acre worth? 2 ofanacre? 3aacres? 25a? acres? 


What is the value of ae O 


WRITTEN EXERCISE 


1. A man had of an acre in a village lot, and bought = 
¢ 


of an acre adjoining. How much was his land then worth 
at d dollars an acre? 


2. What is the value of the land in Ex. 1,if a=1, d = 2, 
ec = 300, d= 400? (Simply substitute in the result.) 


3. One automobile can go m miles in f# hours, and 
another can go : as fast. What is the rate of the second 
one per hour? 


4, What is the value of the result in Ex. 3, if m = 63, 
I reap Des CS SS 


5. If m pupils are divided equally in ¢ classes, and if half 
are boys, how many boys in each class? 


11. 
i 
15. 
18. 
a1. 
522, 
23. 


24, 


MULTIPLICATION OF FRACTIONS Tl 


WRITTEN EXERCISE 



































ab ce dt 9 (DE CO28 6): 
bc d at " @d ef? ab? 

mn pg 4 Ue eenee W -1 

co in ns? pes wa ez 
a+b « xy*z . a—b ab mint 

wy yz 2 “mn mn? 8 
Old (a0 00°C: : (meee Ce aec Oo 
Aab’c® 5 pq?r? eh Sai ae 
ae ee 10 17 a°b'c® §=259 min? 
Ty Ra ATI " 37m?n 119 a"! 
ie mene pane ge Re Oem Ey 
DT Te Te iG, i i ae ae 
LI ac MATA GIO yoke aoc ga 
Sh Thee aie ie TT LY z 
Se ieee et L 1 —-1 
= 16. a%-— 19, — 5 

mn mn n? ie 7 

1 if —1 -—-1 1 1 


abcd? a*be*d DOA a 007% Cay ap ay 


Go DD. 6 0 OO. 


ii eod oanbte. 87h 





p+atrts rs py —7s 


py pg 7st 8 


e—2ey+y? ays — abc 
bei ihe — abe xyz 





ee Ot 


abe LYZ iat 


12 DIVISION OF FRACTIONS 


DIVISION OF FRACTIONS 
ORAL EXERCISE 


1. Because 2.3 = 6, we know that 6 + 2 = how many? 
Because a:b = ab, we know that ab + a = how many? © 


2. Because — in = = at we know that at > 5 = = how many? 
90. Division of fractions. — Because = = = e and also 
noe Js 4 we see that 
wa 6 a 


The result of dividing by a fraction is the same as that of 
multiplying by the aa enverted. 


me _m y 

Bet to divide = Wee —, we may mul- 7 7 e ao. 
tiply ~ by the result being — pds ee 
Nx 


First indicate the work and then cancel if possible, as follows: 
pease IESG OS EV oo 83) 


@ pt Sp @-pt-6q 2p? 


WRITTEN EXERCISE 

















Th Ged “q_. mn —a’be — bed? 
ib, eee ee 2 ee Sy Sepa e 
be 6? mn pq? x ye Yz 
Cpa | eile en ee Stabe: = bea? 
4. oe Oe Oe Sy reer gma 
c ab Py m: pqr pqr 
Eee ene pe 
gd ek ' 8m 2 xy? sais 
a(a—26) _ a®b a? +h?  m* 
2 2b 4o Mera. Se 
iL —mn —n?*m 12. a+ 2ab + UF ab. 








Lp) {Her ab a 


DIVISION OF FRACTIONS | 13 











Ci Cometc meen?) m® 
13. ---+-——: —— 5 —— 
De Tithe Si wm n 
ab be | abe mn np  mnp 
Beeta Pe de 16g mg 
e ace. abc? " aryrz? ale? abe 
UE! Be ho Be Cee Cyan 7/2 
x). Pg 3 
gps 
ee Ve ald oes 
ee (; ‘) Bez Re eae 8 8° ab? 
93 able ah a o4 27 a*b?c? 119 m?n? _3 
ab Cae UG pd AW SES SY 
25. see Ve YF, 26 tag! aL 
Ui Ps 8 6 TN OE py py 
apy ares ai A 98. (ett vir) 1 
abed 4 Pq gr par 
(me Ce ke \OLUG facta 
ge (; c+$) Me. 4 
30 vy —ye ww, — ww 
rae ut ay ye 
31 ey te Pod Oo 
; ry ye ape 
re Vee 0 Ce 
Semen mG ge cog ne 


33. One train travels m miles in f# hours; another, d 
miles in ¢ hours. The first rate is how many times the 
second? 

34. One man earns d dollars in w weeks, and another 
earns m dollars in ¢ weeks. The weekly wages of the first 
are how many times those of the second? 


14 LINEAR EQUATIONS 


LINEAR EQUATIONS 


91. Various kinds of equations. —There are many kinds 
of equations. The equation «+2=5 is very simple; 
2%—7=6 is more difficult; 27+ 7x2=18 is still more 
difficult. 

92. Linear equation. — An equation like 2 x — 7 = 6, in 
which the unknown quantity has no exponent except 1, is 
called a linear equation or a simple equation. 

The name linear is the more common in advanced work, but 
both names are used. The expression x! means the same as 2. 


93. Solution of an equation. — To find the value of the 
unknown quantity is to solve the equation. 

94. Axiom. — A statement assumed to be true is called 
an axiom. 

95. Axioms needed in algebra. — The following include the 
axioms already used (page 4) in solving equations. 

Axiom 1. Jf equals are added to equals, the sums are 
equal. 

Thatiis, 1f ¢ — 3.— 7, then 7 = 7 + 30rd). 

Axiom 2. If equals are subtracted from equals, the remain- 
ders are equal. 

That 1s,1f ¢ + 3 = 9, then z = 9 — 3, or 6G. 

Axiom 3. If equals are multiplied by equals, the products 
are equal. 

That is, jf 1a = 5, then 7 = 4X 5, or 20. 

Axiom 4. If equals are divided by equals, the quotients 
are equal. 

Lhatis;i1i0 ¢ = 39, theiereeo 0 1, Orne 

Axiom 5. Quantities which are equal to the same quan- 
tity, or to equal quantities, are equal to each other. 


EQUATIONS ies) 


ORAL EXERCISE 
1. Solve the equation 2 +2=5; alsor+2=08. Isb 
considered as known or as unknown? (See page 14, § 16.) 
2. Solve the equationa +4=6; alsox+a=06. 


Solve the following equations, finding the value of x: 
3. en = 4,.%—n=™M. 5. 3a = 6. 
62.0 Gea. rE Cay Te See Onis. 
See = 14g. 105 na 3 i. Lio 4 2 a7, 


iA Se TR atten, 1B yon eek ies WE BE Pm hans eh 


WRITTEN EXERCISE 


1. Solve lle+F=7 +46. 


moOlvert ae — oe oe + 21. 
. Solve 2e¢+3e+7=224 33. 


se 5 — 3000. Find the value of x: 


2 
3 
4, 
5. Solve 3(2 « + 200)= 1200. (First use Axiom 4.) 
6. Find the value of « in the equation 52+10=32+4+ 32. 
7. Find the number whose fifth and seventh together 
make 24. 

8. Find the number whose half, third, and fourth 
together make 13. 

9. Find the number which added to 4 equals 7 more 
than half the number. 

10. A man saved half of his wages for 5 years. He 
then had saved $3000. What were his annual wages ? 

11. A man by saving $200 more than } of his wages _ 
annually for 3 years, saved $1200. What were his annual 
wages ? : 


} 


76 LINEAR EQUATIONS 


96. How to solve equations. — We have seen that the solu- 
tion of a linear equation consists in arranging the x’s in 
one member and the known quantities in the other, and 
dividing by the coefficient of a. 


97. Transposition. — The subtracting of terms from both 
sides of an equation so as to carry them from one side to 
the other, changing the sign, is called transposition. 


Thus, in the equation 8 — 2 = 12 — 42, we subtract 3 from 


both sides, leaving —x=12-—427-3. 
We then subtract — 42, or, what is the same thing, add 42, 
and —x+4xr=12 -3, 
or One so. 
whence ioe ts 


In this solution we transposed the 3 and the — 4z. 


98. Solving by transposition. — Since we have now solved | 
so many equations that we can use the word understand- 
ingly, we may say that to solve a linear equation, 


Transpose the terms involving x to the left side, and the 
known terms to the right, and divide by the coefficient of x. 


ites x 
Thus, if — + 5. =~ " 
us, 1 oie rere 


Leenene 1 
then, transposing, 5 = 2, or ge Le 


The coefficient of zis}. We may divide by }, or, what amounts 
to the same thing, multiply by 2, and 
ae 
Check, 2°-4+5=43-447. 


WRITTEN EXERCISE 
1. 42¢%—1=2- 81. 2.1342 — 21 —427 — 3: 
3. 122—-6=92+4 27. * 4.15¢—9=112+439. 
5. To 2 — 32=—162-+ 86. 6. 252+8=162 + 89. 


EQUATIONS {yf 





























(hs 2 Bo path 

9. E42 = 69. 10 nef 11 
M5 +3=249. 12 2FF 13 =48 
13, = _=7 _ 49 14. 5+ 5+5=18 
15, FF Sa 16 22 2 _ 135 
lie 1 9 — bo.) 18) 9 F631 
19, 24° == 31. 20, “= 4 2th 
21. x +10 %x = 605. 22. « +120," = 14.56. 
gg, =F _ ett __o ag, “EE _ Sty 
a5, “Hy ts ag, 4 FSS _ 88, 


27. Find a number whose seventh part minus its eleventh 
part equals 4. 

28. Find the number whose third plus 7 equals the 
number less 3. 

29. Find a number which, when subtracted from 88, 
equals the number less 17. 

30. A man lost 32% of his capital and then had $4896. 
How much had he at first ? 

31. A man gained 15% on his capital and then had 
$8625. How much had he at first? 

32. A man has 27% of his capital invested in a farm. 
The farm is worth $1917. How much is his capital? . 


78 LINEAR EQUATIONS 


33. Find a number such that $ of it less 2 of it equals 19. 


34. What sum increased by 1% of itself ‘amounts to 
$2585.60 ? | 


35. Find a number which increased by 17% of itself, 
and then decreased by 36, equals 198. 


36. A dealer sold some goods for $2802.40, thus making 
a profit of 13%. What did the goods cost him? 


37. A man lost 30% of his library by fire. He had 
630 books left. How many had he before the fire? 


38. A collection agency charges 4% for its services in col- 
lecting a debt, and remits $912. How much did it collect? 


39. The sum of a certain number, a third of the number, 
a fourth of the number, less 7, equals 4,4. What is the 
number ? 


40. A bank charges 0.1% exchange on a draft. The 
entire cost of draft and exchange is $1751.75. What is 
the face of the draft? 


41. An agent charges 5% for collecting rents for Mr. 
Glover. He deducts his commission and remits $332.50. 
How much did he collect? | 

42. Half of the remainder found by subtracting 7 from 
a certain number equals a fourth of the sum of the number 
and 7. What is the number? 


43. An agent bought a building lot for Mr. Roberts, 
charging him 3% commission. Mr. Roberts sent him the 
price of the lot and commission, amounting to $2626.50. 
What did the lot cost? 


44. The income of a certain store increased 163% the 
second year it was open, and the income the third year 
was 25% more than the second year. The income being 
$3500 the third year, what was it the first year ? 


CHAPTER II 


OPERATIONS CONTINUED. FACTORING. PROPORTION. 
EQUATIONS 


MULTIPLICATION 
ORAL EXERCISE 


1. Multiply x by a; by 6. What is the sum? 

2. Multiply m+n by a; by &. The sum is the product 
of m+n by what expression ? 

3. How can you multiply any polynomial by a+ 6? 
(Multiply first by a; then by what term? Then what 
should be done?) 


99. Multiplying by a binomial. — Multiplication by a bino- 
mial in algebra is much like multiplication by a two-figure 
number in arithmetic. 


Arithmetic: Algebra: 
Multiply 45 by 23. Multiply a+ 2b bya+6 
45 Cited 
25 ‘ Oa 20 
135 product by 3 a* + 2ab product by a 
900 « 20 ab + 2 62% eee, 
M55,“ 28 Oo a0 2.02 Sata 


4. Why do you begin at the right to multiply in arith- 
metic? Why may you begin at the left in algebra? Could 
you as easily begin at the right in algebra? Try both plans 


on the blackboard. 
79 


80 MULTIPLICATION 


ORAL EXERCISE 
1. Multiply 4a° by 5a*; —3a* by 6a°; —5a? by — 42%. 
2. State the product of a -—a’?-2a-—38a; also of 
327-—4a4".—2@. 
3. Tell how you proceed to multiply by a+ 6; by 2a—); 
by 4a? — 36°; by any binomial. 


WRITTEN EXERCISE 
Multiply in Evs. 1-20: 


l.a+bdby a+b. 2.a+ybya+y. 

3. m+n? by m+ n?. 4.2a+bby 2a+0. 
5. a—bby a—d. 6.2—ybyx—y. — 

7. 3m—n by 3m—n. Sasa by Ota 
9. a+b by a—b. 10.a—bbya+o. 

ll. x—2ybyx+2y. 12. m?+ 3n by m?—3n. 
13. 2a+36b by a—26. 14. Tx?y?+1 by 2 277?—3. 
15. 627+1 by 52? —3. 16. 4mn+ayby3mn+4ay. 
17.a+b+cbya+b. 18. 77+ 2ay+y* by «+y. 


19. m?—2mn+n? by m—n. 20. 4a?+4ab+0? by 2a+b. 

21. What is the product of 27 and 23? of a+6 and 
a—b? Suppose a = 25 and b= 2? 

22. How many square feet in a square that is 42 ft. ona 
side? How many square feet in one that is Vos t ft. on 
a side? Suppose f=40 and t=2? 

23. If a man earns $27 a week for 27 weeks, how much 
does he earn in all? If he earns ¢+s dollars a week, how 
much does he earn in ¢+s weeks? 

24. The product («@+y) («—y)=2?—y*. Hence write 
down, without multiplying, the following products : (10+ 2) 
(10 — 2); 12.8; (20+ 1)(20 —1); 21-19. 


SQUARE OF A BINOMIAL 81 


ORAL EXERCISE 
1. State the product of a+ 6 by a; by &. Add them. 


2. Write on the board the product of x+ybya+y. 
How are the terms formed from x and y? 


100. Squaring a + b, a — b.— Consider the squares of 
a-_o and of a — 0. 





The square of a + b. The square of a — 6. 

a+ 06 se Say 

Gare 0 Cano 

arts ab product by a a? — ab productby a 
ab + O26 “0 — ab+h « 2 Bap 

Aa GUT * a0 C2 00S Dae « a—b 


101. Square of any binomial. — It is therefore seen that the 
first term of the product is the square of the first term of the 
binomial. The second is twice the product of the two terms 
(negative when one of them is). The third is the square of the 
last term of the binomial. Therefore 

The square of a binomial equals the sum of the square of 
the first term, twice the product of the two terms, and the 
square of the second term. 


3. State the squares of b+c; of d+c;o0f 2+a; of 27+1. 
4. State the squares of a—a; of x—y; of a—2; of a? —-y’. 


WRITTEN EXERCISE 


Write out the squares of the following without multiplying : 


1. p+g@. Pane y ieee 18 Oe ea ty 

4, x7? —t. 5. be — x. 6. 26—d. 

7 ap. 8. 2a+0. 9. mn +1. 
10. mn? + p. 11. ayz —1. 12. por + 2. 


13. 2a+ 36. 14. 3a—20. 15. 5ay+1. 


82 MULTIPLICATION 


ORAL EXERCISE 


State rapidly the squares of the following: 


l. ptr. 20 at 3. 0 +a. 

4.9g—h. Spice SES Ta 6. an 9. 

7. «2 — 7. 8. d? — b?. vas OT a 
10. xyz + 1. 11. xyz — 3. 12ers a 


WRITTEN EXERCISE 


Write out the results without stopping to multiply: 


1. (ab +c). 2. (38 — ¥°). Sue (ied) oo 

4. (2— 2°)? 5. (a+ 5c)? 6. 4(a@+y)”. 

Cate). 8. 1 —7q)’. 1 el (ic bia bak 
LO(pe tL) 11. (8a + 2)?. 12. (6a — b)?. 
13. (@ —11)*. 14. (xy? + 1)? 15. (x —Ty)?. 
16. (2a — b)*. Ling 22) See Bate), 
19. (a+ 7b)? 20. 10—3y)%, 21. (way +2)? 


22. (20% +41). 23. (80x%—1)7. 24. (8m? +4 5)” 
25. (xyz +10)% 26. (abed +1)% 27. 11 +22)%. 
28. 100(@+y)% 29. 2Qu+2y)?, 30. d0x+ 10y)? 


102. Pictures of squares. — In the figure point to the line 
that equals «+ y. Point to the area «?; to 
vad vt an area xy; to another area xy; to y*. What 
then does the square on « + y equal? 


31. Draw a figure showing the square on 
104 2. 
32. Draw a figure showing the square on 10 + 1. 





33. Draw a figure showing the square on 2” + y. 
34. Draw a figure showing the square on 3x + a, 


PRODUCT OF SUM AND DIFFERENCE 83 


103. Product of a + b and a — b. — Another product fre- 
quently met is that of the sum and difference of twe 











quantities. 
a+b @—b 
GD 0 
ie 0 C00 
— ab — b? C0 
ie — } a? — } 


That is, the product of the sum and difference of two 
quantities equals the difference of their squares. 


WRITTEN EXERCISE 


Write the following products without stopping to mul- 
tiply : 


1. («+1)@—1). 2. (1+a)(1—a). 

3. (a? —1) (a? +1). 4. (ab —1) (ab +1). 

5. (abe + 2) (abe — 2). 6. 22+") (2x2 —y?"). 

7. (1+ mx?) (1 — mz’). 8. (p?>+4q)(p?—49). 
9. (8 —42°)(8 + 42°). 10. (Sabe — d) (5abe + d). 


11. 10a?—1)(10a?+1). 12. 2a?+11) (12a? —11). 
13. (Sayz +7) (Sxyz —T). 14. 100m + 3) 100 m — 3). 
15. 42+7)42 —7);19-5. 16. d1 —4) (11+ 4); 7-15. 
17. (xyz + 1)”. 18. (7 —2 xy)’. 19. (327+ 4) 
20. (84 2 pr)’. 21. (8mn—T)*. 22. (2 mn — 3). 
23. (104+ 3mn)*, 24. (aye? 2)?, 25. (0274 3 y)?. 
26. First write the product of the two binomials; then 
square the result: [(a + 0) (a — d)}?. 
27. In the same way write the results of the following: 
[ (m?+1) (m?—1)]?; [(2a—3) (224+3)]?; [(1—-5a)A+5a)/. 


84 MULTIPLICATION 


104. The product of two binomials. — The product of two 
binomials like « + 2 and 2 + 5 is so frequently needed as 
to require attention. Consider two such cases: 








ge +2 a2 —T 

ee) z+3 

ue? +24 a? — 7x 
52+ 10 32 — 21 

a’+T7Ta2+10 g—4y— 21 


ORAL EXERCISE 


1. In the first product how is the 7 (the coefficient of x) 
formed from the 2 and 5? How is the coefficient of x 
formed in the second product? 

2. How is the last term formed in each of the products ? 
Can you now tell the product of «+5 and «+ 5 without 
actually multiplying? © 


105. Absolute term.— In the expression «?+ 7x + 10, 
10 is called the absolute term. 

106. (x + a)(x + b).— Zhe product of x +a and x + b is 
x? plus (a + b)x plus ab. 


WRITTEN EXERCISE 


Write out the products without stopping to multiply: 


1. (0 +7) (x +1). 2. (7@+1)(@—1). © 

3. (a + 10) (a —7). 4. (pg + 8) (pq + 9). 
5. (xy — 3) (ay — 3). 6. (8a + 5) (ba — 5). 
7. (mn — 11) (mn + 1). 8. (mn + 6) (mn + 6). 
9. (p’¢ + 7) (p’¢ + 10). 10. (abc + 6) (abe — 7). 


11. (xyz + 20) (xyz — 5). 12. (xyz + 15) (ayz + 2). 


CHECKS 85 


107. Checks on multiplication. — Because the product of 
a+aand «+6 is «74+(a+6)x+ ab, whatever values 
may be given to 2, a, and 6, we may check our work by 
giving any convenient values to these letters. 

For example, to check (# + 3) (# — 2) = 22 + x — 6. 

Let c=1. Then (1+3)(1—-2)=4--1=~—4, and 2 +1 
— 6 =— 4; so the work is probably correct. 

' Try alsox=5, Then (5+ 38)(5—2)=8-3 = 24, and 52+ 5 
—6=25+5-6= 24. 


108. Ease of checks. —It is usually easier to check the 
work than to look at an answer in a book. 


Work: Chieti ete yi be 
2e¢+ 3y == 5 
do — Ty =— 9 
8x? + 12 xy —15 
—14ay— 217 
8a27— 2aey—21y =—165 


WRITTEN EXERCISE 


Multiply and check: 
1. (23 x + 21) (« — 17). 2. (21a + 37) (9a — 14). 
3. (x? + 15 y) (a? + 14y). 4, (15%? + 7) (17 x? — 8). 
5. (l7x+2y)A5xe—6y). 6. (85%+4 23y) (15% —Ty). 
7. ily—22)(2y—112). 
8. (31 xyz + 2) AT xyz — 7). 
9. (15 a*b?c + 7) (9 ab*c — 8). 
10. (27 xy? + 1) (80 xy? — 3). 
11. (322 + 3y)(— Alx+8y). 
12. (231 a + 147) (3829 a — 176). 
13. (321 a? + 17 6) (151 a — 55). 


86 


14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 
31. 
32. 
33. 
34. 
35. 
36. 
37. 
38. 
39. 
40. 
41. 


MULTIPLICATION 


(a + 0) (a? — ab + 0°). 

(a — b) (a? + ab + 0”). 

(a + b) (a? + 2 ab + 0%). 

(a — b) (a? — 2ab + 0°). 

(a + b) (a — b) (a? + 0°). 

(81 abe — 1) (91 abe + 17). 

(62 abe + 1) (78 abe + 3). 

(2% — 3) (427—Tx-+ 2). 

(4a —1)(82?— 22443). 

(42 a — 37 b) (51a — 170). 

(111 ax +- 21) (97 a*x + 2). 

(4x? —4 ay + y”) (2% — y). 

(9a* + Gay +") (Sa +). 
(a+ 3b) (a+4b)(a+ 50). 

(172 a?y? + 27) (172 ay? — 2%). 
(42 p?q?r? + 8?) (17 p?q?r? — s?). 
(a+ 6) (a+ b)(a+ 6) (a+). 
(Tx + 2y)(9a? +4 xy —6y?"). 
(a*b?c? — 2 abed + d?) (abe — d). 
(ax? + by?) (ax* + abx?y? + by). 
(a + b) (a + 30% + 3 ab? + 0°), 
(a®b’c + 1) (a%b*c? + 2 a®b?c + 1). 
(8% —3y) (Gx? — Tay + 50y?). 
(x — y) (#® — 3 xy + 3axy? — y’). 
(21 x? — 0 (152 a* + 73 x? — 41). 
(— 27 a?y? — 2?) (— 21 x7? — 2). 
(208 —Ty)(9a° + 15 ay + 8 y?). 
(6a + 7b) (36 a + 84 ab + 4907). 


REVIEW 87 


109. Illustrative problem. — At what rate will $320 pro- 
duce $19.20 interest in two years? 


1. Ii the rate for 1 yr. is r%, for 2 yr. it is 27%. 
2. Since 27r% of $320 = $19.20, 


9) 
1 = 4 of 819-20 


$320 Sie 





WRITTEN EXERCISE 


1. What per cent of $75 is $3.75? 
2. What per cent of $15.50 is $0.62? 
3. If x% of $156 is $4.68, find the value of z. 
4. If x% of $730 is $65.70, find the value of x. 
5. What sum increased by 6% of itself equals $1007? 
6. On what sum is $11.25 the interest for 1 year at 
41%? at 5%? at 3%? at 2%? 
7. At what rate of interest will $450 produce $15.75 
in 1 year? in 2 years? in 9 months? 

8. At what rate will $240 produce $28.80 interest in 
2 years? in 3 years? in 4 years? 

9. How long will it take $350 and interest to amount 
to $386.75 at 34%? at 5%? at 4? 

10. At what rate will $260 and interest amount to 
$282.75 in 3} years? in 5 years? in 2} years? 

11. What sum will amount, with interest, to $381.50 in 
Z years at 44%? at 5%? at 6%? at 2%? 

12. A man lends $250 for a year at a certain per cent, 
and the next year he lends the $250 and the first year’s 
interest at the same rate. The sum of principal and 
interest at the end of the first year is $260. What is the 
rate? What is the sum at the end of the second year? 


88 FACTORING 


FACTORING 


ORAL EXERCISE 


1. What are the factors of 15? of ab? of 25? of a?? 

2. Name two factors of 12. Are they prime factors ? 
If not, state the prime factors of 12. 

3. What is the product of a+ 6 and a+b? What 
are the factors of a? ++ 2ab+ 67? of a+ 2axy + 7? 

4. What is the product of «—y and x—y? What 
are the factors of a? —2ay-+ 47? of m*—2mn-+n?? 

5. What is the product of m+n and m—n? What 
are the factors of m? — n?? of p? — q?? of 4 — a?? 


6. What is the product of «+2 and «+3? What 
are the factors of 274+ 54 4.6? of a7b7+ 5 ab +6? 


110. How to factor expressions. — Factoring always depends 
upon remembering types of products. 

111. Monomial factors. — Because we remember that 
x(y + 2)= ay + xz, it follows that the factors of xy + xz 
are candy+z2. That is, 


xy + xz = x(y + 2). 


WRITTEN EXERCISE 


Factor the following : 


laa: 2. am + mb. 3. e7+ 2 xy. 

4. py — qr. 5. a*y + yx. 6. p* — 3p. 

7. abe + bed. 8. ax* + axy. 9. ab+ dbe. 
10. 3pq? + 6¢*. 11. mx? — nxy. 12. c6 + 3c?d?. 
13. 2 x*y + 6 2. 14. abry + wxyz. 


15. may + nxz + que. 16. 32 ay? 4+ 8ayz +4 xyz. 


USES OF FACTORING 89 


ORAL EXERCISE 
5 16 ab 
ib atin OE Ge 
; 29 a(atb) wx(a+1) ab(m+n 
BuPATso the following: at aah aniy ee. 
3. How do you ordinarily reduce fractions to lowest 
terms? Give three illustrations. 


1. Reduce to lowest terms these fractions : 


112. Uses of factoring. — One of the chief applications of 
factoring is in the reduction of fractions to lowest terms, 
so that the fractions may be used more easily. 
e+3nr 
e+4z 
CeO er L(g apt 9 
e+4e x(@t+4) 2th 


= Ts 
to lowest terms. 
q 


to lowest terms. 





For example, to reduce the fraction 











Also, to reduce the fraction 


ie ee a ing AN 





ay—Ty y(e—7T) ¥ 
by factoring and then canceling « — 7 from both terms. 


WRITTEN EXERCISE 


Reduce to lowest terms the following fractions: 











L eae 9°. mas coe 3. an ab 
m* — ™m x“ — 2x a+a 
x + xy m? + mn 427+ 62 
4, ———__+: So oe Uo aecewersapar so 
xy + zx n? + mn 2ay+8u 
2pqa+¢q? 3mx + x 81 — 27y 
‘1 8. ——-———_ 9, ———___-.. 
2pr+ rq y +3 my 63 +18 y 
ae _ 2 22 
i oie CRN PUT pegs 


abe — 7 be ' ga —3 yz wt 3 pq? + par 


90 FACTORING 


ORAL EXERCISE 


1. What is the square of a+ 0? of a—b? 

2. Squarea+y; x—y; 2x+1; 1—3y; ab+cd. 

3. What quantity squared equals a+ 2ab+0?? 2? 
—2Zay+y?? 4¢+4e44+1? at4+22?41? 

4. What are the two equal factors of m*—2m4+1? 
of xy? +2ay+1? a@+t4ab4+4b?? at —2a?+41? 


113. Squares of binomials. — Because we know that (a + y)? 
=2+2aey+y’, and («—y)?=2?—2ay+y’, it follows 
that the factors of these trinomials are known: 

a ree ae pS ae Napa 
x? — 2xy + y® = (x — y)’. 
For example, the factors of 4922+ 14241 are 7x +1 and 


Ve 1 or 
A) to Ae gen ( da) Are ( Cae 


== (1-214). 
So the factors of 9 22 — 12 ay + 4y2 are 3x — 2yand 32 —- 2y. 
For a2 eyed? (0 2) (OD) 2) ee 
a CMF oe PO be 


WRITTEN EXERCISE 


Factor the following : 


Gr 8 gehen Pee 2 a2? + 2aen + n’. 

3. ee 41. 2 ae. 

5, Agee 4 el |e 6, 145744 * 

4. 9276741; 8. 1627 — Sze + 1. ° 

9. 277? + 2 aye + 2 10. a*b? — 2 abed + ¢?d?. 
11. pt? + 20 pg + 100. 12. 6%? +121 + 22 abe. 


13. 9a* + 42 a*l? + 49 4. 14. 36.x* + 60 xy? -+ 25 y*, 


DIFFERENCE OF SQUARES 91 
ORAL EXERCISE 
1. What is the product of a+ banda—b? What are 
the factors of a? — b?? of a? — y?? of m*—n?? 
2. What is the product of 2~7+1and2x2—1? What 
are the factors of 4a?—1? of 4m?—1? of 1—4a?? 


3. What are the factors of the difference of two squares ? 


114. Difference of squares. — Because we know that (a + 7) 
(x — y) =x? — y’, it follows that the factors of the difference 
of the squares of two quantities are the sum and difference 
of the quantities. 

x? — y? = (x + y)(x—y). 
For example, the factors of 25 #4 — 121 y? are5 22+ 11y and 
pe Ly HOT 
De gtee 1Aley ipa (eA TRG yy 
(ae ey) ee hy) 
Similarly, (e2—y)?-4=@-yt+2)@-y-2). 


WRITTEN EXERCISE 


Factor the following : 


1. p?—4¢". 2 2a- = 4b? 

3. 1644-7”. 4. 64 m‘n? — 1. 

diye sake 6. 1 — 100 abc? 

7. (a+ db)? — e’. 8. Zax +a*?+ 27. 

9. 36 a? — 25 0". 10. 144 m? — 25 n?. 
11. 36 p?q* — 121. 12.0 Slt 25 7 07e70 
13a? 4-2 ab +- 07. 14. 25 ab? — 25 b’c?. 
15. va* + be + cry. 16. 4m2n+ n?+ 4 m* 
17/564 a4? — 31 o7d*. 18. a2 = 4 ab + 407. 16: 


19. a? +2ab4 0? —4. 20. 1+6(@+y)+9(@+y)*. 


92 FACTORING 


ORAL EXERCISE 


1. What is the product of ¢+1anda+2? What are 
the factors of 27+ 3a” + 2? 

2. What is the product of x—3anda+4? What are 
the factors of x7 + #—12? of w+ ay —12y?? 

3. Of what expression are x + 5 and « + 4 the factors? 
alsoxz—VTande+44? alsox+l and a—8? 


115. The product of two binomials. — «a + b 
Because we know that (x + a) (x + 6) x 4 a 
=x*+(a+ 6)x + ab, it follows that a bx 
we can often tell the factors of expres- ax + ab 
sions like x? + (a + b)x + ab 


e7-- 7¢ —18 and 27+ ba +6. 
For example, the factors of 2? + 7x—18 are (x+ 9)(x— 2). 
For w+ Te—18= 22+ (9 — 2)\a -- 9(— 2) 
ee ED G3 ta PE 
That is, 9 and —2 are two numbers which added make the 
coefficient of x, and multiplied make the third term. 


WRITTEN EXERCISE 


Factor the expressions in Hrs. 1-14: 


Vea A 2a st eee 10: 

3. 2 —5a4+6. AG ee os 

5. p?>+18yp-+ 81. 653 ga 8, 

1. TRL 2 ee, 8. p* — 20 p + 100. 

9. pg? +13 pq + 36. 10. a*b* + 6 ab — 40. 
11. a?y?z? + 11 wyz + 24. 12. m?n? +17 mn + 72. 
13. mn? — mp* — g?m. 14. (a + b)? —(e + d)*. 


15. Of what expression are ab? — 4cd? and ab? + 17 ed? 
the factors ? 


REDUCTION OF FRACTIONS 


ORAL 


Reduce to lowest terms: 


1. 


a+26 ; 

G4 O 

m+n 

m? — n? 

a(b+ Cc) 

6? — ¢? 

he ae I es 
Apes ite 
oy 

x2—2ey+y 


EXERCISE 





2. aaa 
m(m — 1) 
m= —" 1. 
peeks eee 
"e+ Q2ay + y? 
ae (tary aie 
x?—2aey+y? 
a? — ¥? 


roam 


93 


116. Reduction of fractions. — As usual, in reducing frac- 
tions to lowest terms, jist factor, then cancel common factors. 


WRITTEN EXERCISE 


Reduce to lowest terms: 


or 


10) be 


Coe 2ab+06? 
O70 = ab 
ie re 


x? — 52+ 6. 


ee Oe — 3 


a 00 OF 


pt —8p+12 


ara + ya? + are 
(a + y)c? + cz 
1— 15% + 562° 


1—17 24+ 7227 


a + a —12 
gt ig 220 
a+2a— 8 
a*7t+6a+8 
(4 + 6)*—1- 
eee eee. 
ipa a ea 
72 —— 3 e+ 2 
ge? — 22% + 40 


4m? —n* 


10. 


ag Aon? eed mn 


94 REVIEW 


WRITTEN EXERCISE 
1. Solve the equation 3 «+9 = 5a —65. 
2. What sum increased by 38% of itself amounts to 
$785.89 ? 
3. What sum decreased by 7% of itself is reduced to 
$709.59 ? 


4. The cost of a draft, less the discount at 0.1%, is 
$9740.25. What is the face? 


5. The cost of a draft, including the premium at 0.1%, 
is $10,760.75. What is the face? 


6. A man offered $7790 for a farm, which was 5% less 
than the asking price. What was the asking price? 


7. A man pays an agent $7725 for buying a house, 
which includes the agent’s commission of $225. What 
was the per cent of commission? 


8. A man pays an agent $5610 for buying a house, 
which includes the agent’s commission of 2%. What 
did the agent pay for the house? 


9. A man bought two horses at the same price each. 
He sold the two for $198.90, gaining 11% on one and 10% 
on the other. What did he pay for them? 


10. A man bought two horses at the same price each. 
He sold the two for $163.20, gaining 10% on one and los- 
ing 6% on the other. What did he pay for them? 


11. A man increases his original capital by 7%, and the 
next year he decreases what he then had by 10%. He 
then had $8667. What was his original capital? 


12. A mine increased its income 11% in one year, and 
the next year it increased this new income 10%. The 
income then amounted to $335,775 a year. What was its 
income at first ? 


DIVISION | 95 


DIVISION 
ORAL EXERCISE 

1. Divide 4 a7 by a’; by 6; by 2a. 

2. Divide wa + ab by x; bya+6. Divide m? — mn by m. 

3. What are the factors of a?—2abd406?? Divide 
a*— 2ab+b? by a—bd. 

4. Divide a? — 0? by a—b; by a+b. Divide m?+ 2m 
+1bym+1. Divide a —1 by a?—1. 


117. Dividing by binomials. We have just seen some 
examples of dividing by binomials. Dividing in algebra is 
quite like dividing in arithmetic. 








Arithmetic: Algebra: 
Divide 3772 by 46. Divide 2a?+ 7ab4+ 386? bya+30. 
82 2a te a) Check: 
46) 3772 a+3b)2a4+7a+388 a=1,b=1. 
3680 = 80 times 46 2a7+ 6 ab 12+4=8. 
92 ab + 3 6? 
ag A ab +3 0b? 


118. The work is easier in algebra, however, since we need, 
after once arranging both dividend and divisor in the same order 
as to some letter, to divide only the first term of the dividend by 
the first term of the divisor to find the first term of the quotient. 

Here 2a?=+a=2a; subtracting 2a(a+ 35), there remains 
ab + 3b? to be divided. Then ab +a=b; subtracting b(a+ 3b), 
there remains nothing. 

Therefore 2 a? + 7ab+30b?=(2a+6)(a+3b). Inother words, 
the quotient is 2a + 6. 


119. We check, as in multiplication, by putting numbers 
for letters,ias a—1,b0—1. 


96 DIVISION 


WRITTEN EXERCISE 
Divide in Hxs. 1-15: 
l. w+2ey+y by x+y. 
2. 2m? — mn — 3n? by m +n. 
3. 3x°?4+ ry —2y? by 3u— 2y. 
4. 6p? + pq —2¢@7 by 8p+2¢. 
5. 42a?—13ay+y* by Tx — y. 
6. 2+3a%0+38al?+ by a+b. 
7. 10a? — 29 ab + 100? by 2a — 5b. 
8. 20 p?q?r? + 41 par + 2 by par + 2. 
9. 40 27y? + 58 ay — 21 by 10 xy — 3. 
10. 30 m* — 229 m? + 30 by 15 m? — 2. 
11. 3 —47 ay + 170277? by 1 — 102. 
12. 32 2?y? + 46 xyz — 52? by 162xy — z. 
13. 1 —18 por + 77 p’q?r? by 1 — 11 par. 
14. 50 xty* + 77 w?y?2? + 3 24 by 25 x?y? + 2”. 
15. 3224 — 5027+ 3 by 162? —1; also by 4a +1. 
16. If a body moves uniformly 42? + 3ay — 7? feet in 
4x — y seconds, how far does it move per second? 


17. If 3%+7 pounds ona lever will raise 12 x7+ 19% —21 
pounds, how much will 1 pound raise? How much will 
x +2 pounds raise ? 

18. If «+ y articles cost 3”7+ 2zy — 7”. dollars, how 
much does each cost? Supposing « = 1 and y = 1, what 
is the result ? 


19. If there are 1082? + 144277 + 362zy? words in a 
book of 36x pages, how many words are there, on an aver- 
age, to each page? If there are 32+ lines to a page, 
how many words, on an average, to a line? 


REMAINDERS QT 


ORAL EXERCISE 
1. Divide by x: ax + ba, ax + 0. 
2. Divide by x+y: (@+y), («+ y)? +m. 
3. Divide by « —y:2?—y*?, 2*7—-—2ay+y*,0—-—y—a. 
4. Divide by a+1: (a@+1)’, (a4+1)?—4, (a41)?— 2’. 
120. Remainders in division. — In dividing by binomials 


remainders are treated in the same way as in dividing by 
monomials. That is, they lead to fractions in the quotient. 


For example, divide 








9 a2 
Bry te + oy bye ty. ee roll 
Rearranging the dividend ary 
; 2 » 22 
and dividing in the usual way, x+y) = tery + 3y 
there is a remainder of 2 7. et ry 
Therefore the quotient is ty + 3 y? 
ryt ¥# 
2 a2 esi OE 
Ly 


WRITTEN EXERCISE 
Divide in Hrs. 1-4: 
1. w—2ayt+Ty by«—y; byx+y. 
2.0 +3m—38m by m+1; by m—1. 
8. 8+ 3a%y4+38ay—6ybyx+y; byx—y. 
4. 6a—17a%+4+11ab?4+ 50? by 2a—b; by 2a+0. 
Rearrange the dividend and divide in Hrs. 5-9: 
5. 17 ab + 21 a? + 5 0? by Ta + 6; by Ta— b, 
. 8797 +6 p? +7947 by 6p4+ 9; by 6p—g¢. 
. 17 avy — 137° + 62? by 3u—2y; by 82+ 2y. 
. 6 m+18mn —12n? by 2m+7n; by 2m+5n. 
. 138 mn + 3 n? + 20 m? by 5m%+4+ 2n; by Sm—2n. 


©o © = OD 


98 DIVISION 


121. Arranging terms. —It often happens that in attempt- 
ing to arrange the terms of the dividend in the same order 
as those of the divisor, certain powers will be missing. In 
that case zeros may be inserted if desired. 

For example, to divide x? — y? by x — y either of these forms 


may be taken. After a little the second one will naturally come 
to be used. 














xt2+ xy + y* x +ay + 7? 
a—y)e®+ OF O-% a—y)x — 
a — xy a3 — ay 
Cpa cM) xy — yP 
x2y — xy Ly iy 
xy — yp xy — 


WRITTEN EXERCISE 
Divide in Hxs. 1-6: 


1 2+yby x+y. 2. a—b by a— bd. 
3. at — yt by at+y. 4. 82° —1 by 22 —1. 
5. oe +y by x+y. 6.14 27a by 14 3a. 


In Exs. 7-10 one factor is given in parentheses ; 
required the other factor: 
7. @&+3a?+5a4+3, (a+1). 8 1— 640%), (1 — 4 xy). 
9. 32m> +1, (2m-+1). 10. a°b*c® — d®, (abe — a). 


Since at is equal to a?-a?, at — bt may be thought of as the 
difference of the squares of a? and l?. Hence 


a* — b* = (a? + 6?) (a® — 6?) = (a? + 0?) (a + 6) (a — B). 


Factor the following: 
11. a* — 7%. 12. 16 m* — ni’. 13. 16 p* — 1. 
14.°8lat— 169%) Set 916.81 p494s* = 413 


FRACTIONS Je 


FRACTIONS 
ORAL EXERCISE 


a(a + 6) a(b ae) 
b(a+6) a(e+da) 


to a fraction with the denominator 





1. Reduce to lowest terms: 





2. Reduce au 
x 


(x + y)*; with the denominator x? — y”. 


122. Polynomial denominators. — The method of reducing 
fractions is the same for polynomial as for monomial 


denominators. 


22+1 
For example, to reduce 


EOL ie al nO 
ce eA Viet (dee 2) =, 
2. Therefore. x — 2 must be multiplied by x — 3. 
3. Therefore both terms must be multiplied by « — 3 (§ 73). 
‘ PR eM 3 LGR SATO To Ee VO Alito Shoe Se 
ae iQ (E93) (062 ))  @—5at 6 


to a fraction with the denomina- 





gL 





123. Changing Signs. —It should also be remembered 
that, since we may multiply the terms by —1, we may, 
change the signs in both terms. 


For example, Sty kee 








Qt ti eB 


hod 


WRITTEN EXERCISE 


Reduce to fractions with the denominator x? —9x +14: 











e+.2 yeaa xo? + 6 a? eee ae) 
ee a 27 Ba ee eh iy eee 
With the denominator 6 a3? — 29a? 4+ 46a— 24: 
5 oe A Ole 7 Boe 5 Cereal. 














ee) ee ae a ane ORE 


100 FRACTIONS 


ORAL EXERCISE 


1. Express as mixed numbers: %, $) 72: 
oe he -. 
oy RE a ee 


a a 





2. Reduce to mixed expressions : 


1 . 
A | 





3. Express in fractional form: 2}, a + 7 x + 


124. Mixed quantities reduced to fractions. — Required to 


2 


“a 


as a fraction.* 





a ——= 
express a — b — 
a 





+b 
b 
1. Since 1 pepe 
a+b 
—b b 2 — fp? 
2. Therefore a—-b= & Bs at Ne oF 
a+b at+b 


3. Therefore 
a—-& a@—b? a-— p 
a—b-— = = 
a+b a+b atb 


i 9D Sn) 

















4, — 
a+b 
ie w—-b—-a+h a—a 
o. = ---_—_-_ = . 
ath a+b 


WRITTEN EXERCISE 


Express in fractional form: 








eae. 2. 3%. 3. ate. 
1 | x 
a a Pai 1 3 ; 
4. m “a 5. e+ y+ 5. PEO ae 
6 m, ee ty 
7 Carlen 8 en hey ‘ 





10. Lr ee Bi plat 11. m—1——- 12. a+6— 


13. 


15. 


Whe 


19. 


21. 


23." 


25. 


27. 


29. 


31. 


33. 


35. 


37. 


39. 


41. 


43. 


























REDUCTION 
a+ 6? a. 14. 
=o 
1 
one 
x wep 16. 
2 
{en eae A 18. 
Deeg ae TZ 20. 
a— 
a 
ipa aon er am Pepe. 
5 p-l1 
np? — 24 
Merde etsy 
a? — 73 
2 2 é 
Gielm I Sey, 26 
4 x? 
EPR INT RIESE: 28 
Wo 2 
mt-n— 30 
m—n 
hs wits 
2h Pe 2 ee 
a?+ 6* + c7+ ara 32 
272 
tatgepeec ee ag 
tab 
ere et el 
8x7? —3ay y+ 36 
—4¥P 
er+2ary+ cee . 38 
Ni Ra dees LS 
2p? — 5 nq 
Z29+3 q+ ———: 40 
pP q peg 
Dy ale yee 
Dip ch ype ea 
3m—dn 
32a — 7b? + 60? 4+ 4. 44 


101 


ta ti—=. 





5 ae or. 
ow+i— rr ae Es 
2 q° on 
Pe Han rcay, 
8 
2 pass ab, ie, & 
oe aie 3 p? 


tat+l——+at 











Te Da 
RUE SEE = reas 
~- 6m+21 
UP an Gs aoe 
ep re} 2 
i ele 
ax —b 
a + 2 nity" 
Be Bs 0 bg voy 
3 2 
are te alg OE 
G0 
2m? 
ie ts oe i i 
. 1d Tm ais] 
ae on 
.m+m*+m men 
eh 
Ee aee Reyne ee — 
Leo 2 
20-4305 ——", 
he 
Bey Aye ae 


a—b 


. 91a? —62b? + 310?— 62d?, 


102 


FRACTIONS 


ORAL EXERCISE 


Reduce to entire or mixed quantities : 
































1 32 ab 48 m?n 3 x — 9? 
" 160 "12 mn oy 
7. (a+ b)* 4a? — be . ot") 
a+b Za—b ' 25 (a — y) 
7 a*—2ayty” 3 at(@e+y)* (4 + b)* +m 
; xv—Yy a+b 
21 2» 
125. Illustrative problem. — Reduce 2 r a as : to an 
entire or a mixed quantity. : 
Since a fraction is an expression of division, 
v+3ae4+9 if 
WRITTEN EXERCISE 
Reduce to entire or mixed quantities: 
“L a — ues 9. at + De 
ety ato 
CT Pes MO 
3. . 4. : 
ea p— 2 
z e+ 3a+y? F a*—92+6 
a+ty x —4 
: op aettshie ans eiViameea Us LO: 
; ZT f ay —1 
e+ta+ta+ 1 Tet ae OO 
a 10. =? __ 
a+l pg —11 
ey? + 11 2y 4+ 28 mn*p? — 23 mnp + 60 
Li . 12. —?—_——__—___—__. 
ey +7 mnp — 20 
ee a> — b eh a> + On 14. e+ 3sx°%y+3ay?+y 
Gab a+b 


e+y 


REDUCTION 1038 


ORAL EXERCISE 


Reduce to fractions having the lowest common denomi- 
nator : 


1. 4, 4. 2. 3, ¢. 3. 8, t. 
rh ae a ec i) 
oh aN Seat rar 


~ and —~— to frac- 
be cp + cq 


tions having the lowest common denominator. 


126. Illustrative problem. — Reduce 


1. Since the factors of the denominators are b, c, p + q, the 
Pe-dsis 0¢(p.-h.d). 





etter ar a a(p + q) 
2. Multiplying the terms of — b + q, we have ——————. 
plying be Bey | be(p +9) 
bx 
3. Multiplying the terms of 2 by b, we have ah ea 
cp + cq bc(p + q) 


WRITTEN EXERCISE 


Reduce to fractions having the lowest common denom- 
nator : 























ee 9 Iho ee 3 ee 
Cget "m m+i1  Qpr—r? 7? 
a b aa t 1 mt-1i 1 
4. aes 2 HY, 3 ae 6. Si ei Te pers 
ah as imal UY a’—a a” m+m m 
a b m iy x z 
« be’ ae +e? cae mn +n? gs ye y(e +y) 
ry 2 4 1 
i, SR ER i ee eM So ee Oe ree 
Deen? 2 PY PY-—P Och IEP AY CH 
13 we eay + yy ett tay ty" 4 m—1 m 
tay” vv Gite i SL 


15. Solve the equation 14% —175 = 172 — 241. 


104 FRACTIONS 


ORAL EXERCISE 


Reduce to lowest terms: 











REE . a Za+b . 3. atb- 
6 way 4a* — b? a” — b? 

4 x+y 5 4(a+b) 6. Wawa 
“e+ 2ey+y? 8(a+6)?  . m* —1 
aye(a + y +2)? © x.— Y ve (2(a+y)* 


 eyet(aty+te) #2 —LZey+y? ~~ 36(a@+y)? 


127. Illustrative problem. — Reduce the fraction 
v?+8a+15 


ete) eee be 
to lowest terms. 





et Oo ti Lore bs) (Leto en at : 
oe . 1G 
eee) irk SAS (Mire ON 4 ater (5) eda ad (38 ) 


WRITTEN EXERCISE 


Reduce to lowest terms: 


ie ge tae erm Ltt yo 16 48) 


" 2? + 20a + 99 a y2+19y + 84 
3 Pe lop 20 4 m* +18m +17. 
' p? +18 p+ 65 mm? +19m + 34 
5. ot 16 800 3 py tA py —T7 
x? + 14a —120 py +2 pq —99 
7 xy? +8 xy — 48 8 Orne 00 
 2?y? + Say — 84 | ' 2? — 17 ay — 847? 
9 mn? — 26 mn + 105, 10 a*b*c? — 15 abe +- 26 
nee mi 076767 18 abe + 65 


pg? —18 par $17 7? 12 mn? + 20 mn + 91 


ok pq? — 19 par + 847? m?n? + 21 mn + 104. 


REDUCTION 105 


WRITTEN EXERCISE 


Reduce to fractional forms: 

















1 bl we ++. 2. id casch SA aie oe 
ee ee ioe mnt 
eae ae m—n pr nee on eae 
me Pe ee ee 
5. 8 p*q? er aT GMO ty ey VST 
17—2? m—n 
oy dee Wey FA : 1 ae ROE doy eee 
te Lay? — 17 ary ers 8. 23m IED 23m 17 
Reduce to entire or mixed quantities: 
gl + gilt — 3 yi t+ 3y't+2y? 
Ls ar pa © LO encase aay a 
m4 Py + 1) 
x? + 22ey+121 xy? — wy® + 2 
ee 12. 
ee 11. xyS +1 
xt +4 a8y + 42%? 2 
13.2 ee eee Ue 14. jee ANG ea ee 
e+ y (ie 
22 ; = 
15. mn 17 mn + 15° ys ut — $4a7y? + 289 ys 
mn — 5 x manier 


Reduce to fractions having the lowest common denom- 














inator : 
y Core el i ins 
ih SS aero Ch eae ee 
eee de — 4 
19. Bary eer he 90. Lt38y Lys 
xy — yp xy 3x2 -+6y Gay 
91 api 1 dpsed: 99. aes © i ar 8 din i 
' @b?— ab ab? 2 (p—9q)~ 4 
meee PT 4, ot 2 abe oes Oe 


Ang? + q° : 2q° an— be ar 


106 FRACTIONS 


ORAL EXERCISE 


1. Add ; and a3 — and =; and *. 
2. Add 5 and = and 1. 


“— and t cea a oa 
a+b GSI as e+td 
4. In adding fractions, what must be the nature of the 


denominators ? 














3. Add 


128. Addition of fractions. — Required to add the fractions 
a b 


en ee 


1. The least common denominator is evidently ) (b + c). 
| Bos (OGG) Genel see ae) 
DO Gere) 0 ime ere 
b b? b? 
bte (b+e)o B+be 


ae ag 
3. Therefore the sum _ ab tact 
b2 + be 


9 


me 








and 





WRITTEN EXERCISE 














1 a ee 
Le gaye a oan 
peer Pee 
m+mn mm ola 
Le Slee 0 m 1 
2 au ie ab 2 n' men 
gS ala Oa eae ee P78 
“b(a+b) a+b “p(p+ay p 


9. Solve the equation 3% +17 =z + 59. 


ADDITION OF FRACTIONS 


107 


10. A man bought some goods at 6% discount from the 
What was the marked price ? 


marked price, paying $1128. 


Ly 
13. 
15. 


17. 


Solve the equations in Kxs. 


21. 


23. 


25. 
27. 
29. 


31. 


1G 


9p? + 6 pg 


og 39 

Pon 
x x 
m2 | m? + 4m? 
x—y Le 


Wy —= Ui 


Aad 
oe 


mtn 
m? 


ad, 





3p 


x 4 4 
—~+———+ y 
9 Suede 120. 


x 46 
4e44+7=22+41. 
32 —x = 48 — 132. 


Get a 


182% —175 = 157 x. 


ee) 
o 5 











mn? + 2 mn 
ge 


as 


12. 


14. 


16. 


18. 


20. 


21- 


24. 


28. 
30. 


be iI 
ab zs ac 





abe 
4° aoe o+e 


x+y 


Bie 


cee 


ek 





-{- 


Me eae 
act ra ee (a — 
3 ie ye 5 





vel GPE) eek te a 


x 
ye ao + 141. 
Gils (te A at 
ee atte 








6) (622 a = 98 9) 
20. 
ee 


Ala —76=>17x2-+ 
19.43 « — 83.6 = 
Ze 


32. lr a ee ee 


33. Find the number whose half, third, fourth, and sixth 
together equal 15. 


34. What number diminished by 3 equals the sum of the 


half and fifth of the number? 


35. Find the number whose half, fourth, eighth, 
sixteenth together equal the number less 2. 


and 


36. A man’s salary when increased 8% amounted to 


$1350 a year. 


What was it before the increase ? 


108 


129. Subtraction of fractions. — Required, from 


tract 


a— 


b 
a+b 





FRACTIONS 


a 
=~ b- 
;, to su 


1. The least common denominator is evidently b(a + 6). 
a (a+ b)a a? + ab 


9 


ae 


4. The difference = 





b (a+b)b ab+b 


Ci Dy mb ( isbn bia 
a+b b(a+b) ate 











a+ abi— (ab — 6?) 














CL 





a 4b 


ab+ 6b? 
er ab — ab mine Met 
je” ab + b2 ~ ab+ b? 
WRITTEN EXERCISE 
at i 
2. —— 
Faery 
“an  a—w7 
4,.—— . 
by b—y 
hee Bhi 
6: a+ 0 ah 
b 
a—b 
8. 


(ea 
“@ @+y 
Digs te ONL 
By cea 

1S Re. 
P+tY pr 
a b 1 
be ca abe 


Pad Saas 


ame 


. Solve the equation 


. Solve the equation 5 a + : ae 
) 


ue 
a 


Aseitges 


ees 


Sy 
4 
. Solve the equation 2” + 8? = 163. 


. Solve the equation 3 x — 7, « = 7007. 


ath 


(a+b)? “(a-+5)* 
ei+a— 1 ieee 


atl 


+ 10 — 





a 


Lice 
h 





SUBTRACTION OF FRACTIONS 109 


15. A man borrowed a sum of money for 1 yr. at 5%. 
The sum of principal and interest was $262.50. How 


much did he borrow ? 








i ea ee 
yte y 

2 

ie, Se 

[Die ee 


a y 








1 ie begs BS 
TSG T ane 
a4 nls 
he ee ee ey a 
DOD 
rat a—2ab a 


27. 


pf 


g 


"pt +Tp 


3 
aa 
P 
hy 2 
qe 
a b 
Ly + 2u xy 
oe a 
ary is: Cay ie 


a b? 





. 
2 





€ 


C972 — 12a m? 


Solve the equations in Hxs. 28-35: 


28. 5— 5 = 5. 

30. i cae 
32. 2-2-7 +2 
Se | 
34. = — 30=7-1 


eo 
a 


oS WIS WIS WIS 


& 


“| 
~] 


36. What number decreased by 14% of itself equals 172 ? 
37. What number increased by 8% of itself equals 37.8 ? 
38. What number increased by 10% of itself equals 


687.50? 


39. If from a fourth of a certain number I take a seventh 
of the number, the result is 3. What is the number? 


110 FRACTIONS 


ORAL EXERCISE 


Uy ele ae Chin 
5° 


- of ot 
il a ,x+4 1 1 
pa sae Pe cal 
Gens a+b ) a Ome 
a i L+Y coe a 
4. How do you multiply a fraction by an integer? a frac- 
tion by a fraction? 
5. In multiplying fractions why should you first indicate 
the multiplication, then factor and cancel, and finally actu- 
ally multiply ? 


ik 
1. How much is 5 of 3? = of 7? 3 Oboe 











2. How much is : of 


3. How muchis Bx? a: 9 (0-7-0) ae 


130. Multiplication of fractions. — oe ool the last ques- 
— ab x 


a piacere 





tion (Ex. 5) with the product ote 


az— ab x = atx? — abxz? 


—————-.= to be reduced. - 
m2 + xy a® ata? + ary’ 





If we first indicate the multiplication, and then factor as far 
as possible and cancel, we have 

a(a—b)z* (a—6b)z 

a(at+y)e (x+y)a 





, a reduction easily performed. 


WRITTEN EXERCISE 











a ba + by 9 Spy ey (e+ Y). 
"6b ax — ay ' 5 wyz 9p 
»n a+b 2m? na — n*y 
31 asa eas. oe n® 4m? + 6m? 
atdiea 10a—15 3mn —4 
2: ee se, pat Tn 10) 3mn +4 


DIVISION OF FRACTIONS eal 


ORAL EXERCISE 


mer. 1 1 
1. Divide = by 2; =, PY 83 pret SY. 





2: Divide 1 by 33 2 by 33 a by 53 je -  DY 


MS = 1b 
a 1 2 a se 
Be Dine b by 3 iby 2 cniby a ieby 5 ye by pe 


4. Divide 5 by 2 by 3 ~ by - How do you divide 


one fraction by another ? 


131. Division of fractions. — We have already learned 
that there is a short method of dividing one fraction by 
another. Just as we divide 3 ft. by 2 in. by reducing to 
the same denominator (3 ft. + 2 in. = 36 in. + 2 in. = 18), 
so we may divide 3 by 2 by reducing to the same denomi- 
nation (3 + 2= % + 8=9+8= 2). But it is easier to 

Cad (ad 


a 
see that 3-2 — 3-3 — 2, and Da eta ap a as 








already explained on page 72. 


.., 40—8a' 16 ab? 
Thus, to divide 623 — ay Vv 15 ay 
toring and multiplying by the reciprocal of the divisor, 
15 xy -4 a? (a — 2b) a Sxy(a— 2b) 
16 a*b?.3a°(2xe—s8y) 4a°b (2x —3y) 


» we have, by fac- 


WRITTEN EXERCISE 











Gg SR ye ie amd alee ay eat A 

Pe ot x PU ir 
ey a a+b ab abl 

2) ea OS ease ean 


112 EQUATIONS INVOLVING FRACTIONS 


EQUATIONS INVOLVING FRACTIONS 


ORAL EXERCISE 


1. By what numbers could you multiply } and have the 
25 om 

Be Tp varac ie 
2. What is the smallest number by which we can mul- 





product an integer? also 


? 





tiply 5 = to have the product an integer? also 5 z ; 


Jefe 5 = 7, what is the value of x? By what number 
do we multiply both members of the equation? 


4, If 2 = 6, what is the value of x? Also find the value 





of w if — eepleme 2 Mik Giger pa se ame 


132. Illustrative problem. — Solve the equation : +7=9. 


Ab Since > +7 19; 


2. Therefore : = 2, by Axiom 3. (State it.) 
3. Therefore x = 10, by Axiom 4. (State it.) 


WRITTEN EXERCISE 


Solve the equations in Hus. 1-9: 


te eee ae ot 32 
lotsa i. 2. i a RE opae ak —6=3. 
4.2¢4+7=15. 5. 2%4+6= 42. 6. 32—7 = 83. 
7 2a+7=}a 4 23. 8. §x—6=>3¢% — 5. 


9. If from 3 of the number in our class I take 9, the 


result is half of the class. How many have we? 


CLEARING OF FRACTIONS 113 


QRAL EXERCISE 


1. Ifi«=T7, what does x equal? 
2. If 2¢@=8, what does 2x equal? a? 
3. If 34 = 2, what does 3x equal? «? 


By what should we multiply both members in order that 
there shall be no fractions in the following equations ? 


be |) 2m 2a 4 2 
4. 7 Tae aaa oS ae ages Ao pt 

rs eae ea ee x 3 
Race tra i me i a pO a a 


8. In general, by what should we multiply both mem- 
bers of any equation in order that there shall be no 
fractions in the resulting equation? 


133. Clearing of fractions. — Multiplying both members of 
an equation by such a number as to have no fractions in 
the result is called clearing an equation of fractions. 

To clear an equation of fractions, multiply both members by 
the least common multiple of the denominators. 


134. Illustrative problem. — Solve the equation 
34+ 24=42 4 2. 


1. Clearing of fractions by Axiom 3, multiplying by 15, 
45 + 10¢ = 12% + 30. 
2. Subtracting both 45 and 1242, so as to place all the z’s on 
one side and all the known terms on the other, 
102 — 122 = 30 — 45, 
or —i2¢=— 15. 
3. Dividing by — 2, to find the value of 2, 
ge == 48, or 7. 


Check. 3 + 2-18 = $-15 + 2; for each equals 8. 


y 
~ 


4 EQUATIONS INVOLVING FRACTIONS 


WRITTEN EXERCISE 


Solve the equations in Kars. 1-10: 





fipekes ee 2, oe 
URIS aS oy 1 aL 

a ora, aioe aera 
es eae i Abie ail x 

5, We G. 7a ao 

6.04 0 eA, 8. da+4xe=3%+4 }. 


9.10+0lea=5+1e% 10. 01¢24+6.2=0.3% + 0.2. 

11. Find a number whose half, third, and fourth added 
together equals 36. 

12. A man spends every year $200 more than half his sal- 
ary. In 3 years he saves $900. How much is his salary ? 

13. After selling $ of his farm, and then 4 of what was 
left, a man still had 140 acres. How many acres had he 
at first ? | 

14. There is a certain number such that its fourth added 
to its fifth equals one less than its half. What is the 
number ? 


15. A man invests half of a certain sum at 6% interest, 
and the other half at 5%. The total interest for a year is 
$66. How much did he invest ? 


Solve the equations in Exs. 16-21: 





x x x LOWE RLD 
16.75 —jg=3 t+ Eee Re rece 
gee ao es they 
2 4 x x x 
22 x x x x 
20 a a7 2 20. 21 6tyo77p7U- 


DIRECTIONS FOR SOLVING 


135. Illustrative problems. —1. Solve the equation 
etd 
aN 


1. Clearing of fractions by multiplying both members by z — 5, 


ko 


pe 


2. Two numbers have the ratio of 2 to 3. 
tracted from the larger, the result is 7 more than the 


LS: 


x 473 = l.8 2 — 9. 


. Subtracting 1.8 z and 3 from both members, or transposing, 


—0.8%= — 12. 


Dividing both numbers by — 0.8, 


smaller. What are the numbers? 


in 


ax “ ° 
1. Because aes ta: we may conveniently take 2x and 32 to 
aes 


represent the numbers. 


oo Then 
and 


22+7= 


8. Therefore 3a2—3 = 
4. Therefore 32 — 22 = 


ing), or 


5. But we wish 2 x and 


3x2 —3 = the larger less 3, 
the smaller plus 7. 


7 + 3, by Axioms 1 and 2 (transpos- 
ea ht) 

3 x2, and these are 20 and 30. 
Check. 30 — 3 = 27 = 20>+ 7. 


WRITTEN EXERCISE 


Solve the equations in Hrs. 1-6: 











ees. 

“zt4 6 

5 

ei 

9 —z2z 
eet 

a 3a 3 





ee lea ye 
ge lara houses 
120 am 
4. Tae amou 
6. —2_ 4+2 


Frsebe sub: 


116 EQUATIONS INVOLVING FRACTIONS 


136. General directions. — We have now found the general 
plan of solving a linear equation: 


1. Clear of fractions. 

2. Transpose the x’s to the left side and the known terms 
to the right. 

3. Collect the terms and divide by the coefficient of x. 


For example, solve the following : 








2 » 
ie “4+ -=+7=8. 
UMD aH 
It will evidently save some work to transpose the 7 first. 
2. Then = poe! 
G0 aa 


3. Clearing of fractions by multiplying by a(a — 6), 
ais 2axr—2 bz = ala —D), 
. Simplifying and collecting terms, 
(3a —2b)x=a(a—b). 
5. Dividing by the coefficient of 2, 
pee RS oN) 4 eee 
3a—2b ad — 20 


use 


WRITTEN EXERCISE 


Solve the equations in Hus. 1-8: 























1 te+7=32 + 18. Eee eas 
3 Saya oat A Bee Be, 
; 5 ti eas 6 ote tam 
7 <4" 41 4+a=0 8 mdse =32—1 


9. A flag pole is so broken off by the wind that } of the 
part broken off equals } of the part left standing. The 
original height was 90 ft. How much was broken off ? 


PROBLEMS oe 


137. Illustrative problem. — Rob can gather the apples on 
his father’s trees in 6 hours, and Tom can do it in 5 hours. 
Working at the same rates, how long will it take the two 
together to gather them ? 


1. Rob can gather 4 of them in 1 hour, and 
Tom can gather + of them in 1 hour. 


2. If it takes the two together x hours, they can gather : 
of them in 1 hour. 7 


3. Therefore Le a 
Cia) mee 
4. Therefore 5z + 62 = 30, by clearing of fractions, 
or LL on 
x= 2;%. 


Therefore it takes the two together 2,5, hours. 


Solve the equations in Hxs. 1-6: 

















Piel Peds tey <i. Ce Dee Oe 
oreo Lee 2 of GRY Ae 
wipe Tea Gata oe Dec 
Rid Fp ey age GaP Rig seep 
pee eNE | Bboy teen er ek ae Lae ee 
“EE Re Ge te oe ee 


7. The sum of two numbers is 90, and one is 20% less 
than the other. What are the numbers? 

8. A piece of cloth lost 20% of its length in shrinking, 
and was then 60 yd. long. How long was it originally? 

9. The perimeter of a certain rectangle is 200 inches, 
and the length is half as much again as the width. 
Required the dimensions. 

10. A man sold a pony for } more than it cost, and the 

buyer sold it for } less than he paid for it, receiving $60. 
How much did it cost the first man? 


118 EQUATIONS INVOLVING FRACTIONS 


11. The sum of two numbers is 50, and the less i is 2 the 
greater. What are the numbers ? 


12. If 110 be divided by 1 less than a certain number, the 
result is 362. What is the number? 


13. If 175 be divided by 3 less than five times a certain 
number, the result is 834. What is the number? 


14. If to jj, of the sum of 3 and three times a certain 
number there be added 4 of the number, the result is 5. 
What is the number? 

15. A man spent $150 more than half of his income 
each year for two years. He saved $800 in the two years. 
What was his annual income? 

16. The distance around a certain rectangular field is 
78 rd., and the length is 2.9 times the width. What is 
the width? the length? the area? 

17. The profit to be divided among three partners is 
$10,000. B receives 24 times as much as A, and A 
receives 80% as much as C. How much does each receive ? 


18. The perimeter of a triangle is 8.1 in.; the shortest 
side is } the longest one, and the longest one is 334% 
longer chen the third side. What is the length of each 
side? 

19. Albany is 1 of the way from New York to Buffalo, 
Rochester is # of the way from Albany to Buffalo, and 
it is 60 miles from Rochester to Buffalo. Required the 
distance from New York to Buffalo. 

20. A man started in business with a certain capital. 
He gained $1000 the first year, lost half of what he then 
had the second year, and gained $2000 the third year. 
He then found that he had the same amount with which 
he started. How much was it? 


PROPORTION 119 


PROPORTION 


138. Ratio. — The relation of one quantity to another of 
the same kind, as expressed by division, is called the ratio 
of the first to the second. 

139. Proportion. — An expression of the equality of two 
ratios is called a proportion. 


The ratio of 2 in. to 5 in. may be indicated thus: 2 in. : 5 in., or 
2 in. 





rege The equality of this ratio to the ratio of 4 ct. to 10 ct. may be 
5 in. 

oe: 
indicated thus: 2 in.:5in. = 4 ct.:10 ct., or ~~ = eos 


Dans 10 ct. 








140. A proportion is an equation.— The proportion «x: 2 


LAN es, ; 
= 14:4 may be written : = Wee simple or linear equation. 


Therefore a proportion is merely an equation containing 
fractions, and is best solved like an equation. 


141. Extremes and means. —In a proportion, the first 
and last terms are called the extremes, and the second and 
third the means. 


Thus, in the proportion z:a =): c, x and c are the extremes 
and a and b the means. 


WRITTEN EXERCISE 


1. Write the proportion «: a = 0: c in fractional form. 

2. Using this fractional form, clear of fractions and show 
tiainec ==_ab. 

3. In the same way, using the fractional form, show 


that x = oe 
C 


4. From Ex. 2, write out a statement concerning the 
product of the extremes equaling some other product. 


120 PROPORTION 


142. Relation of extremes to means. —QOn the preceding 
page it was proved in Exs, 2 and 3 that, in any proportion, 


1. The product of the extremes equals the product of the 
means. 

2. The product of the means divided by one extreme equals 
the other extreme. 


143. Terms may be considered abstract. —In the propor- 
tion 2 in.:3in. =5 ct.: 74 ct., it is of course impossible 
to multiply inches and cents together. But because 

Rs 7 4 5 ct. 


Aen EY an mae he =2) we see that 
. gq Ct. z 


The terms of a proportion may all be considered as abstract. 








144. Formerly, before the equation was common in the school, 
proportions were solved by rule 2, above. 


145. Illustrative problems. —1. Solve the proportion 





2:9 = 34: 153. 
* bk ea OR 
1. Writing this Reimers ee have a simple equation. 
5 
2. Therefore yo a8 =2 
133 
17 
2. Solve the proportion 12: 4% = 27: 18. 
3 
1. That is, seh 
Tog Lome 


bo 


. Clearing of fractions, 24 = 32, and 8 =z. 


WRITTEN EXERCISE 


Solve the following : 

1 ot 905. PARSE lt SS i a 
NP ewe ha ef 4. 245:75 = 98: x. 
5. A9¢ bo == eee G6 clo 25 2620 


PROBLEMS 121 


146. Lever and fulcrum.—If we take a yardstick and 
balance it, as shown in the picture, with weights at X and 
Y, we have a lever. The 
point F' is called the ful- Pane Se 
crum. 

147. Law of the lever. — Now if we put F 25 in. from X, 
it will be 11 in. from Y, because the stick is 36 in. long. 
We then find that the weight « at X will have to the weight 
y at Y the ratio 11:25; that is, we a or 

y 25 

The weight at X is to the weight at Y as the distance of F 

From Y is to its distance from X. 


This law, easily proved in class, is called the law of the lever. 
For example, if a 6-ft. lever has a ful- 


crum 1 ft. from the weight W, and if pis : eenere es eee wees I 


the power we must use to lift w, we know F 


that “ = that is, p=  w. Hence to lift 100 lb. we need 
5 


P 
exert a power of only 20 lb. 


WRITTEN EXERCISE 


1. Where must we place the fulcrum under a 12-ft. plank 
that a 56-lb. boy may balance one who weighs 112 lb. ? 

2. A father puts the fulcrum 2 ft. from his end of a 10-ft. 
plank and just balances his son, who weighs 40 lb: How 
much does the father weigh? 

3. Two boys balance a seesaw, the plank being 12 ft. 
long. The fulcrum is 5 ft. from the heavier boy, who 
weighs 105 lb. How much does the other weigh? 

4. If Rob has an iron bar 43 ft. long, and wishes to pry 
~ up a 300-lb. rock, and he weighs 80 lb., how far from the 
stone must he place the fulcrum, making no allowance for 
taking hold of the bar or reaching under the stone ? 


122 PROPORTION 


148. Illustrative problem. — If 6 sheep cost $30, how much 
do 8 sheep cost at the same rate? 


1. Because the ratio of the cost equals the ratio of the number, 
then, if « = the number of dollars paid for 8 sheep, 


2 x: $30 = 8 sheep: 6 sheep, 5 
or eo) ==18 26, 
a 8 
or aes eS ek, 
30 «6 
3. Therefore = me =) a0 40: 


4. Therefore the 8 sheep cost $40. 


Teachers will find it much better to encourage pupils to put the 
unknown term first, as in other equations, instead of last. 


WRITTEN EXERCISE 


1. If 9 yd. of carpet cost $14.40, how much will 7 yd. of 
the same carpet cost? 


2. If I walk 2 mi. in 50 min., how long will it take me 
to walk 23 mi. at the same rate? 

3. If a train travels 105 mi. in 2} hr., how far will it 
travel in 1} hr. at the same rate? 


4. A wheel 2 ft. in diameter has a circumference of 62 ft. 
What is the circumference of a wheel 5 ft. in diameter ? 


5. A -wheel having a circumference of 10 ft. has a radius 
of 133 ft. What is the radius of a wheel 7 ft. in circum- 
ference ? 


6. A man timed the express train that he was on and 
found that it made 2 mi. in 2 min. 8 sec. It was then 
going at what rate per hour? 

7. If the work ona tunnel progresses at the rate of 16 ft. 
a week, and the tunnel is to be 1520 ft. long, how many 
weeks will it take to complete it? 


INVERSE RATIO 123 


ORAL EXERCISE 


1. If it takes you 4 min. to clean a blackboard, how long 
will it take you to clean half of it ? 


2. If it takes you 4 min. to do the work, how long will 
it take two of the class at the same rate? 


3. If it takes 2 men 14 days to excavate a cellar, how 
long will it take twice as many men? 


_ 149. Varying inversely. — In these examples, as we double 
the number of men we divide the time by 2. That is, the 
time decreases in the same ratio that the workmen increase, 
or the time varies inversely as the men. 


150. Illustrative problem. — If it takes 20 days for 15 men 
to do a piece of work, how long will it take 10 men to do it ? 


1. The ratio of the times is a 


2. Because 10 men do it in # days and 15 men in 20 days, 
5 
, and the inverse ratio is io 





therefore the ratio of the men is ae 


ad 





3. Therefore ae = — 
20 10 
; DOS Losers ase 
4, Therefore z= i0 = 30, and it takes 10 men 380 days. 


WRITTEN EXERCISE 


1. I have enough grain to last my 2 horses 3 months. If 
I buy another horse, how long will the grain last? 


2. With a certain quantity of wool I can make 42 yd. of 
goods 24 in. wide. How many yards can I make if it is 
1 yd. wide? 

3. It takes 56 yd. of 27-in. carpet for my parlor. How 
much will be needed if I use carpet 1 yd. wide, assuming 
that it will cut as economically ? 


124 PROPORTION 


WRITTEN EXERCISE 


1. If 17 horses cost $1870, how much will 11 horses 
cost at the same rate? 

2. If 35 men can do a job of paving in 8 days, how long 
will it take 56 men at the same rate? 


3. How long would it take 50 head of cattle to eat the 
same amount of fodder that 50 head eat in 45 days? 


4. If the interest on a certain sum is $112.50 for 1 yr. 
6 mo., how much is it for 2 yr. 3 mo. at the same rate? 


5. An automobile is timed, and found to make 450 yd. 
in 30 sec. At this rate, how long will it take it to go a 
mile? 

6. If 150 mi. of railroad cost $3,600,000, how much 
more will it cost to extend the road 36 mi. farther at the 
same rate ? 


7. It costs me 75 ct. a night to hght my store by elec- 
tricity when the lights burn 5 hours. How much will it 
cost when they burn 7 hours? 


8. It is estimated that it will cost $185,000 to put a 
state road through a hilly part of the country for a dis- 
tance of 37 miles. How much will it cost for 150 miles 
at the same rate? 


9. There were two pieces of sodding, of the same size, 
to be done in our park. It took 3 men 5 days to do the 
first piece. How long will it take the 2 men whom we 
now have to do the second? 


10. A builder had 4 plasterers at work 12 days in plas- 
tering one of the stories of an apartment house. How long 
will it take 6 men to plaster the next story? If the builder 
puts 8 men at work on the next story, how long will it take 
them? The rooms are the same for all stories. 


SIMILAR FIGURES 125 


151. Similar figures.— Figures which have exactly the 
same shape are called similar figures. 


For example, two circles are similar figures; also two squares, 
two equilateral triangles, two cubes, or two spheres. 


ORAL EXERCISE 


1. What is the area of a square 2 in. on a side? also of 
one that is twice as long? (Draw the pictures if necessary. ) 

2. The diagonal of a square 1 in. on a side is 1.4 in. 
How long is the diagonal of a square 2 in. on aside? (If 
in doubt, measure it.) 

3. What is the volume of a cube that is 1 in. on an edge? 
also of one that is 2 in. on anedge? (Build the latter of 
inch cubes if necessary. ) 

4. If one side of an equilateral triangle is 3 in., what is 
the perimeter of another equilateral triangle that is twice 
as high? three times as high? 


152. Proportion related to similar figures. — We may infer 
from the above exercise that, in similar figures, 

1. Corresponding lines are proportional. 

That is, if the radius of one circle is twice that of another, the 
circumference of the one is twice that of the other. 

2. Areas are proportional to the squares of corresponding 
lines. 

That is, if one equilateral triangle is twice as high as another, 
the area is 22, or 4, times that of the other. 

3. Volumes are proportional to the cubes of corresponding 
lines. 


That is, if the radius of one sphere is twice that of another, the 
volume of the one is 2°, or 8, times that of the other. 


126 PROPORTION 


153. Illustrative problem. —If the area of a circle is 3.8 
sq. in., what is the area of a circle of twice the diameter? 

1. Since the areas are proportional to the squares of correspond- 
ing lines, 


Se totem ial 
or ae 
yeh OR! 

2. Multiplying by 3.8, x= 15.2 


3. Therefore the area is 15.2 sq. in. 


WRITTEN EXERCISE 


1. A cylindrical can holds a pint. How much will a simi- 
lar one hold if it is 1} times as high? 

2. A box 3 in. long has a volume of 10.5 cu. in. What 
is the volume of a box of the same shape, 4 in. long? 

3. A toy balloon 6 in. in diameter has a volume 113} cu. 
in. If it is inflated to 7 in. in diameter, what will be the 
volume ? 

4. A triangle has its sides 3 in., 4 in., 5 in. Another 
triangle of the same shape has its shortest side 2in. What 
are the lengths of the other sides? 

5. Of the two triangles in Ex. 4, the first has an area of 
6 sq. in. What is the area of the second ? 

6. A certain projectile for a man-of-war’s gun weighs 
250 lb. What is the weight of a similar one of which the 
length is 10% more? 

7. Of the two projectiles mentioned in Ex. 6, the area of 
the surface of the second is how many times that of the 
first? 

8. A photograph in which a house appears as 1.7 in. high, 
and a tree as 1.5 in. high, is enlarged so that the house 
appears as 2.1 in. high. How high does the tree appear? 


PROPORTIONAL PARTS 127 


154. Proportional parts. — The profits of a business are to 
be divided between two partners in the ratio of 3:5. The 
profits for a year are $4624. What is the share of each? 


Peele x = the number of dollars in the smaller share, 
4624 -—-x= «6 ae “6s larger “ 
ee Lhéeretore * 7: (4624 — 4) =.3 5, 
Bee eine ek 
4624-—a4 5 


3. Clearing of fractions, 52 =3-4624 —3r. 
4, Adding 3 z to these equals, 


8a2=3-4624. 
5. # =3-578 = 1734, the number of dollars in the smaller share. 
6. 4624—1734=2890,° « “ oe eens ea larger 2: « 


155. While this furnishes a good exercise in proportion, teachers 
will naturally encouraze pupils to say that out of 8 parts, 3 go to 
one and 6 to the other. Hence one gets 3 and the other 3 of the 
$4624. The treatment of partitive proportion as a distinct subject is 
unnecessary. 


WRITTEN EXERCISE 


1. Divide $375 in the ratio 2: 3. 

2. Divide $2619 in the ratio 7 : 2. 

3. Divide 1638 ft. in the ratio 5:13. 

4. Divide $4837 between two partners in the ratio 3: 4. 


5. An alloy contains 19 parts copper to 11 parts tin. 
How many pounds of each in 570 lb. of the alloy? 

6. A certain powder contains 2 parts of saltpeter to 1 of 
charcoal and sulphur. How many pounds of saltpeter in 
462 lb. of powder? 

7. Air contains 21 parts of oxygen to 79 parts of nitro- 
gen. How many cubic feet of each in a schoolroom 22 ft. 
long, 18 ft. wide, and 12 ft. high? 


128 PROPORTION 


WRITTEN EXERCISE 


» lf S42 2 = 51% 27 ondsthepvaluesoter 

o Jf 76213377, tind *thesval ueeoeece 

» £623 Ols=754: a simd the valustor 2 

Lig.) l= eos, find the, value ore. 

» Li = 2: bs, tind) thervalue-o1ew: 

. What number has to 5.81 the ratio of 5 to 7? 

. To what number has 15.2 the ratio of 1.9 to 2? 

. Divide $1025 into two parts in the ratio of 11 to 30. 


9. If the surface of one sphere is 7.2 sq. 1n., what is 
the surface of another sphere of twice the radius? 


OIA TF WD 


10. If the area of one equilateral triangle is 8.2 sq. in., 
what is the area of another one with 4 times the perimeter ? 

11. If the interest on a certain sum for 1 yr. 2 mo. 10 da. 
is $21.50, what is the interest on the same sum at the 
same rate for 6 mo.? 


12. The sum of two numbers is 30, and the ratio of one 
to the other is 2:3. What are the numbers? (Let 2 
equal one number.) 


13. Four partners divide the year’s profits, $12,250, so 
that A’s share is 25% of D’s, D’s is twice B’s, and (’s is 
as much as A’s and B’s together. Find each share. 


14. Three partners divide the year’s profits, $6000, so 
that the first receives $1 to the second’s $1.50, and the 
second receives $1 to the third’s $2.334. How much does 
each receive ? 


15. Five boys go fishing. They catch 80 fish, A and C 
catching the same number, and B catching 2 as many as C. 
D catches as many as A and B together, and E catches as 
many as B and D together. How many does each catch? 


SQUARE ROOT 129 


SQUARE ROOT 


156. Square root. —If a number or an algebraic expres- 
sion has two equal factors, one of these factors is called its 
square root. 

For example, because 4 = 2-2, therefore 2 is a square root 
of 4; and because 4 also equals — 2 - — 2, therefore — 2 is also a 
square root of 4. 

Also, because a? + 2ab + 62=(a + b)*, therefore a+ 6 is a 
square root of a?+ 2ab+b?. In the same way —(a + )) is also 
a square root. 


157. Law of signs. — Because + a:+a=a?, and —a:—ua 
=a’, therefore a? has two square roots, +a and — a. 
That is, 

The square root of a quantity is ether positive or negative. 

158. Symbols. — The square root of a? is indicated by Va?. 


Hence V4 = 2, V16 =4. But since a square root has two 
signs, the double sign +, read “plus or minus” or “ positive or 
negative,” is used, thus: + V9 =+ 3. 


159. Illustrative problem. — What is the square root of 
144? 








1. We see that the factors of 144 are 2)144 
9.9 929..3 3. 2)72 

2. Separating these into two equal groups, we see that 2)36 
144 = 2-2-8 x 2-2-8 =12 x 12. 2)18 

3. Therefore the square roots are + 12 and — 12. Ie 


"WRITTEN EXERCISE 


Find the square roots of the following : | 
1. 625. 2. 576. 3. 225. 4. 441. hae (29: 
6. 121. fo AETES 8. 289. Gy tall. 10. 1024. 


130 SQUARE ROOT 


160. Monomials. — Of course the square root of a mono- 
mial square can easily be found. 


For example, because Of it =a", 
therefore Vat = a. 
Likewise Vatbix? = a%B8z. 


161. Trinomials. — Because (a + 6)? = a?+ 2 ab + 67, there- 
fore we can easily find the square root of any expression of 
this form. 

For example, because a? — 6ab + 9b? = a? + 2a(-—36)+(—36)? 
= (a — 3b) (a — 3b), therefore Vat — 6ab+9R=a—38b. 


ORAL EXERCISE 


State the square roots of the following : 


LP abc". 2 Saad ce 

4. 9a*d*. 5. 16:27", 6. 25 a®btx*. 
«. 49 a7b7e?, 8. 64 x*y?. Shs ol ire 
LO LOU 777 ze lin t2iae 12. 144 x?y?z?. 


WRITTEN EXERCISE 


State the square roots of the following: 


1. 274+62+4 9. 29s 657 at 

3. 4a*+ 4a? +1. Amn 

5. 9974+ 627 + 1. 63x — 14a 49. 

Bae Ete AA Gs ee) 8) 407 4 cbc 67 

9. m?+18m + 81. 10. x*°+ 2 x7y? + 77. 
11. 9m? + 6mn + n2. 12. p? + 10 py + 25 g?. 
13. «? —12 xy + 36 7”. 14. 2 — 16 ary + 64 7. 


\ 





15. What is the side of a square whose area is a? + 2 ab \ | 


+ 6?? a —2ab+ 67? 27 + 2a 1? 


SQUARE ROOT OF NUMBERS 131 


162. The square root of numbers. — Required the square 
root of 2209. 


If we let f= the found part of the root at any time, 
and n = the next figure of the root, 
then (f+ n)* =f? + 2 fn + -n4, 

Therefore if we take away f? we shall have 2 fn + n?, and if 
we divide this by 2 f we shall find nearly n. 


Since the entire explanation of square root depends on this fact, 
teachers are advised to see that it is clearly understood, both from 
the figure and from the formula, before proceeding. 


47 =f +n, the root 
2209 contains f2 + 2 fn + n2 





2 — 1600 
27 = 30 609 Bs: 2fn+ n? 
2f+tn=s8i 609 = me Ai 


fo heen 


The greatest square of 10’s in 2209 is 1600. 
This is f?, and therefore f= 40. 
Then 609 contains 2 fn + n?, because f? has been subtracted. 
Dividing this by 2 f, we shall have nearly n. Hence n = 7. 
But 2 f+ n multiplied by n equals 2 fn + n?. Therefore we have 
taken f? + 2 fn + n? = 402+ 2 x 40 x7 + 7? =47? from 2209. 
Therefore 47 is the square root of 2209. 


See also the figure, where it appears iia 1600 
that the large square equals the sum 2 fn = Li OPW) 
of 402+ 2 x 40 x 7+ 77. POT de re 

If we separate the number into a= 49 
periods of two figures each, beginning 2209 


at the decimal point, we shall find the 
number of integral places in the root, but it is not necessary. 


WRITTEN EXERCISE 


Find the square root of the numbers in Exs. 1-8: 
bs 7h Pp Rae OBE BO UU. 4. 5041. 5. 6241. 


1382 SQUARE ROOT 


163. Square root with decimals. — Required the square 
root of 151.29. | 


The greatest square of 10’s in 151.29 is 100. This is f?, and 





therefore f= 10. 12.3 
Then 51.29 contains 2 fn + n?. (Why 151.29 

is this ?) 100. 
Dividing by 2, f = 20, we find n = 2. 2 f = 20 31.29 
We have now found f+n=12, the 5 Paco ap 44. 

square being 100 + 44 = 144. nD f= 24 7 99 
Since 12 has been found, let us call 2 f + 2 Bore 7.99 


this f (for found). Of course this is 
not the same as the first number found; it is larger, because we 
have found more. 
7.29 contains 2 fn + n?, because we have subtracted f? = 144. 
Dividing by 2 f= 24, we find n = 0.3. 
2f+n multiplied by n equals 2 fn + n*, the rest of the square. 


WRITTEN EXERCISE 


Extract the square roots in Exs. 1-17: 


1. 80.4609. 2. 1944.81. 3. 1.1025. 
4. 0.117649. 5. 0.822649. 6. 0.2809. 
In Exs. 7-9 find first the greatest square of 100’s. 
7. 12,321. 8. 54,756. 9. 63,001. 
In Exs. 10-12 find first the greatest square of 1000s. 
10. 21,224,449. 11. 49,112,064. 12. 96,275,344. 
In Exs. 13-17 carry the root to two decimal places only. 
13. 2. 14. 5. Lbeaie SG oe eye 


18. What is the value of (})?? Vi? V4? V4089? 
19. Write out a rule for finding the square root of a 


4 
common fraction. Apply it to finding the value of | pone a . 


QUADRATIC EQUATIONS 133 


QUADRATIC EQUATIONS 


164. Quadratic equation. — An equation containing the sec- 
ond, but no higher, power of the unknown quantity is called 
a quadratic equation. 

165. Complete quadratic. — The equation x? + ax+6=0 
is a complete quadratic equation. 

166. Incomplete quadratic.— The equation x” + 6 = 0, lack- 
ing the first power of the unknown quantity, is called an 
incomplete quadratic equation. 

The terms affected quadratic for the complete and pure quad- 


ratic for the incomplete equation are still used, but are being 
discarded by the better writers. 


167. Axiom 6. Like roots of equal quantities are equal. 
168. Illustrative problem. — Solve the equation «? — 39 
a1 30), 


1. Adding 39, 72°— 169. 
2. Extracting the square root, x=+ 13, by axiom 6. 
Check, (+ 13)? — 89 = 169'= 39 = 130. 


WRITTEN EXERCISE 


Solve the equation } 2? = 72. 

Solve the equation 3 «? — 7 = 140. 

What is the side of a square whose area is 225 sq. in.? 
What is the value of x if x? added to 60 equals 256? 

. Solve the equation 5a? —Tx+75 = 300 —7Tr+ 422, 


6. Find the number which multiplied by the next higher 
number equals 144 increased by the number. 


Bit Pah ae Rola 


7. Find the number which multiplied by the next lower 
number equals 169 diminished by the number. 


134 QUADRATIC EQUATIONS 


169. Illustrative problem. —Solve the equation «(16x — 7) 
= 7 (2800 — x). 


1. Performing the multiplications, 
1647 —7T2=7-2800 —Fz, 


2. Adding 7 z, 1672=— 7 | 2800; 
3. Dividing by 16, Wei d eh Oe deo 
4, Extracting the square root, 


E=E1-5=+95. 


Check. 35(16-35 — 7) = 7 (2800 — 385), 
for 19355 = 19355. 


WRITTEN EXERCISE 


1.-Solve the equation (x + 6) (a — 6)= 864. 

2. Solve the equation 2(10 + x)(# — 10) = 1000. 

3. Solve the equation 4a(1 + x)= 2496 + 4(x +41). 

4. In # seconds an object will fall 162? ft. How long 
will it take an object to fall 400 ft.? 

5. A library is 1} times as long as it is wide, and its 
area is 150 sq. ft. What are the dimensions? 

6. The width of a schoolroom is 90% of its length. 
The area is 360 sq. ft. What are the dimensions? 

7. The area of a circle equals 3} times the square of 
its radius. What is the radius of a circle whose area is 
154 sq. in.? 

8. In the same way find the radii of circles of areas as 
follows: 616 sq. ft., 1386 sq.-in., 2464 sq. ft. 

9. In the same way find the diameters of circles of areas 
as follows: 5544 sq. in., 3850 sq. ft., 49,896 sq. in. 

10. What is that number which multiplied by the number 
just below it and also by the next greater number equals 
399 times itself ? 


PROBLEMS 135 


Solve the equations in Exs. 11-19: 
600 680 

















i, ee 2 ea +7. 
13. 1%, of (nN) =. M4. ST eee 
17. — 13 = 1000 +(@ + 10). 


18. }) (x + 51) («x — 51)= 4624. 

19. 3 of (@ — 1)= 1860 +(# + 1). 

20. What number equals 14,641 times its own reciprocal? 

21. What number has the same ratio to 3721 that 1 has 
to 5041 times the number? 

22. What is that number which, increased by 1, equals 
the ratio of 960 to 1 less than the number? 

23. A field is 3 times as long as it is wide, and its area 
is 2883 sq. rd. ‘What are its dimensions? 

24, A triangle whose base equals ‘its height has an area 
of 10,082 sq. in. What is the length of the base? 

25. A triangle whose base equals twice its height has an 
area of 5041 sq. in. Required the base and height. 

26. A rectangular field is 4 times as long as it is wide, 
and its area is 2500 sq. rd. What are its dimensions? 

27. The base of a certain triangle is 4% more than the 
altitude. The area is 74.88 sq. in. Required the base and 
altitude. 


28. The surface of a sphere is 124 times the square of its 
radius. What is the radius of a sphere whose surface has 
an area of 616 sq. in.? 


CHAPTER III 


FRACTIONS CONTINUED. ROOTS. SIMULTANEOUS EQUA- 
TIONS. THE COMPLETE QUADRATIC 


FRACTIONS 


170. Addition of fractions. — The operation of adding 
and subtracting fractions, as already studied ($$ 83-84, 


128-129), may now be extended. 
Required to add 77 ond eel 
Casta) 1 trees dD) 


Reducing to the l.c.d., we have 


a —2Qry+ 7? 2xy x* + y? 








x* — y? ey? x? — y? 


WRITTEN EXERCISE 








1 a*h? re ab “LY LY 
“(a+b v@—P (e—y («—y/y 
2a 1 par if 
eesti lao Oy ¢ pyr—1' prt 1 
P LW Ye on xy ae Ua 
ayz+yzw wte " atyt — ab? ay? + ab 
4 y eget Stel 3 16n 4 
Sk ee piace t a) a ’ 9m?—16n?  38m—4n 
2_ 9 8 Soe 
9, Hee UT ee a ee ee 
x?+2ey+ty “ty a— 6  @—2ab+ 6? 
iL 1 2 ae 12. LYz 1 











156 


ey ety ae ax ay + az bx + by + bz. 


SUBTRACTION OF FRACTIONS 137 


171. Subtraction of fractions. — From ae Se to 
os e4+2ey+y 
subtract J 














ee. ye 
Say aI a tt 
w+ 2ay ty (x+y)? 
oe ey ie. xy 
ey —@ty@-y 
3. The l.c.d. is evidently (x + y)?(x — y), and we have 
@a-y? ty (ety) _ Pay +  — wy — 2? 
G@+ye-y @+Me-Y CtVH*E-D) 


The denominator may be left factored. It is usually better to 


factor all expressions with which we may be working, so as to 
cancel when possible. 


WRITTEN EXERCISE 











age tial pede 






































oN en =. 
3 3 6a 4 Des LO ENS ee L 
eee 9) eee eee 
fie eh Oo ee f jie A eee 
+ a+b at—b ‘m—n? m—n 
7 ab ah 8 se Te 
: az — b? Reedy i pg? +1 1+ p*? 
$ a+b ab 10 Liste Wet La te 
"V+ &@— 0 Sy oa? — 1 
(2a+b)b a(a—2b) Col ee ee 
oS 12 =e 
b+a a—b a—-1 2#—22+1 
> /) 12h) 
ny, ~ 1S Pe 
pt+tgatr pr+pg+pr Ape Le ab 1 
15 a—b ath 4u+y xy 





e+ 2ab+? a—2ab+o? 2 6a+2y° Q2y+4a 


(138 FRACTIONS 


172. Multiplication of fractions. — Required to multiply 
x — y? 2a+y 
4e°+4ay+y x—y 
Factoring, indicating the multiplication, and canceling, 


(@ty@—y:Crty_ zty 
(Aaa) ay) 2n+y 


WRITTEN EXERCISE 














2 yty 9 m* mn +n 

Ura ere “2? mn —m 
Hei GS Grea ea 4 

3. emma: 4 . . 
(ete (CS set) a*t+2Z2ba atd 

5 9x*—-y? x—38y A ConA 


o—9y? 38a+y ; 2m+1 4(m?—m)+1 


ah ed* ce a 














" Bd we ad b 

“1 at + 2 0%? + bt ase. 
tei a ee 

9 Ma Le tnd OE on ered 
em, = leg eM. 4 

10. a*h + ab +b a 
tt a Sra a ea, 

1 TRACI Shel tg YG, a—2b 


Se ee 
a bse b2—¢ ab 

b+e 0b a? b—e 

16 m®— 2577 10 m?n? 


20 m8n? 4m? + 5 n? 

















ee Se Se ene 
. . . 


DIVISION OF FRACTIONS 139 


173. Division of fractions. — Required to divide 


-a+db 2 ab 


Qa y a? — b2 








Multiplying by the reciprocal of the divisor, as explained in 
§ 90, we have 


G0 (G0) (a Pen (ait) 






































a—b 2 ab 2 ab 
WRITTEN EXERCISE 
L Cie ee 16 aty®—-9 | day — 3 
ab ah? , 2uy A a7y? 
¢ OB NG ee ie et 4 Olt Pte mT 
" ab ' a —$? “m+ 8m+7 > m+7 
F ag ee Pe aN Pe ke Z a +9e+20  2+5 
Sa ae eens eg 0 CES ga 
CR 5 5 Parva, ett or oe 
ptd | pP—_q "  e@+5 § #?+6x+5 
Oe ee a 
eae + et? 
ie atb+e .(atb+e? 
“a—b—e- (a—b—ec) 
rr mn? + 3mn +2 : mn Le 
mn + 5 " mn? — 25 
1 Ee Yo ty ty eee UA 
a? — 2ey + yf? 229 
13 Cg ony, Oey aes 
; at — y' “ety? 
ath at+tb\) . fa—b a+b 
se: (S ee ett) (ott 


140 


174. Illustrative problem. — Solve the equation 


3 
te, 


FRACTIONAL EQUATIONS 


FRACTIONAL EQUATIONS 





12 ae 


see ee: 


1. Multiplying both members by (x — 8) (# + 1), we have 
Otol +8 (4 5 Bor 

2 4% —8= 12. 

3. Therefore 4x2 = 20. 

4. Therefore Ei 0 


Check. Substituting 5 for x, }+%= 432. 


WRITTEN EXERCISE 


Solve the following: 


1. 


OL 





























1 a ak 
E— Nea 
7 2 ff 
baie ye ant, 
1 1 9 
6 a+1. «x(@+1) 
20a eel ee EL 
e—1l-“#+1 27—1 
220 Sh a Rag Os 
@.-- lake leet 
3 1 6 
Don 1 ae Ore orem Teme 
Page, Se 
etl x-—5 2—4e¢—5 
3 20 — 30 








2 b> pees = oe tOs 


3a —4(x—1})=9 —1(5a —7). 


23. 


24. 


PROBLEMS 141 






































22 1—2 
x 3 = 4 — 5 
12 % 48 = 4 
sO oy Se Oe 
eee il tae 
o— ox 2(a—1) 8 
2 3  4x2+7 
oe, en x — 4 
dee wit aoe: 4 24 — oe 
4. 6 3 12 
Eee Cae ey 

















Z2x2+a 32 — a 13° 




















S 5 4. 
CIEE eet EE 
' 5a+2(a —1) 19 ieee? 
%+-35 . 32 a= 
1 + 5 +2+ 5 e=a()) 
9 9 CZ 


a®—Qa+1 24+2a4+1 (@—1)* 


142 


25. 





26. 














Sx+5 
14 
1+ 72 
a—1l 
pei Sa) 

: 3 











ee bane 


FRACTIONAL EQUATIONS 

















Bee 

eo Ree: 
3—T% 42+6 
ieee 


= 0. 











2(1 + 22) 

3(3 — &) 

_204+22) 
3 


ox 
»(02- 


we) 





Heist eaves eyo 


1) -Ee@tsy=0 


37. A third of the sum of a certain number and 5, plus 
a fifth of the difference found by taking 5 from the num- 
ber, equals 6. What is the number ? ; 


38. If 1 less than a certain number is divided by 1 more 
than the number, the quotient is the number divided by 1 


less than itself. 


What is the number ? 


SIMULTANEOUS EQUATIONS 148 


SIMULTANEOUS EQUATIONS 
: ORAL EXERCISE 


1. Given the two equations 7+ 3y=7,x+y=83; sub- 
tract member for member, the second from the first. What 
is the value of y in the result? | 

2. Given the equations e+7y=10,%7+4y=1; how 
may we find an equation containing only y? Find it and 
solve for y. 


3. Given the equations x + 9y =19, «+ 7y=18; what 
is the value of y? How will you now find the value of «? 


175. Simultaneous equations. — Equations which have the 
same values for the unknown quantities are called simul- 
taneous equations. 

For example, a+2y=12,%+y=7, give by subtraction the 
equation y= 5. But if y= 5, we may put 5 in place of y in 
either equation and find the value of x. If we do this, we find 
that 2. 


176. Illustrative problems. —1. Find two numbers such 
that the sum of the first and 10 times the second is 21, and 
the sum of the first and 3 times the second is 7. 


1. Let x = the first number and y = the second. 


2. Then Hig ME Be 929 i 
Pio Y=. 
3. Therefore 7y = 14, by subtracting, 
and Tees 


4. Substituting this value of y in the first equation, 
geo) = 21, and yeas 
Check. Substitute both values in the second equation, and 
1+ 6 


144 SIMULTANEOUS EQUATIONS 


2. Solve the equations 27+ 7y=39,a+y=T. 


If we subtract at once, the x will not disappear. Hence we 
multiply both members of the second equation by 2. (We might 
have multiplied by 7, the y disappearing by subtracting.) 


al. 2 ey ae 
2x+2y= 14, the second multiplied by 2. 
5y = 25, subtracting. 
y= 9. 
x +5 = 7, substituting 5 for y in the second. 
9 


ot Bm oo tO 


. r= 
Check. Substituting in the first equation, 2-2 + 7-5 = 39, for 
4+ 35 = 39. | 


WRITTEN EXERCISE 


Solve the following : 


lee yy Ss 2.2+5y=8, 
ey =D, x—. y=0. 
32s O ia aL, 4,.%— y=3, 
eo = Dy = 9. “x+5y = 15. 
be — 1 Yo, 6. «+4y= 24, 
x+2y=12. e+ 2y=14. 
7. 38a+4y=7, 8. 3%+2y = 20, 
x—y=0. x—-2y=A4. 
OU Die-) Yi ls 10. 42%— y=81, 
e413 Yas Loe e+ Ty=15. 
1l. 374+ y= 40, 12. 2% —) jee) 30, 
x—-2y=A4., x+3y = 38d. 
13. 5e+ 2y = 68, 14. 7% + 6y= 57, 
x+3y = 37. x—9y =18. 
15. 14% —29y =1, 16. 80x%+ 4y= — 34, 


C— (ya), eles AW Toe 1G 


PROBLEMS 145 


177. Illustrative problem. — Solve the equations 
dx—2Zy= 24, 244+ T7Ty=41. 


Since the coefficients of z are 3 and 2, we may make them alike 
by multiplying both members of the first equation by 2, and both 


members of the second by 3. Then 
Le 6x— 4y = 48, 
2 Ga 2by = 1235. 
3. 25 y = 75, subtracting (1) from (2). 
4. y = 3, dividing by 25. 
5. Hence 3x — 6 = 24, by substituting in the first, 
and ie eae Na vel 


Check. Substituting in the other equation, 20 + 21 = 41. 


WRITTEN EXERCISE 


Solve the following: 


1. 4¢%4+ 3y=17, 2.5x%—2Zy=8, 


Darel Ye oe. oa@+ 5y = 42. 
3. 5a—3sy=8, 4.7Tx—d5y=46, 
oSxa+ y=s0. 5x+9y = 58. 
5. da — Sy = 38, 6. 82+ 56y=0, 
Tx —9y= O4. Tx— y=d50. 
i294 --* 27 = 66, 8. 8e+i11ly=— 98, 
Oa Liye O14 3a 2y = 0. 
9. lix—8y=1, LOPS eas ye 90, 
Tx+5y=—T7O0. Sipe toe Wate LOO: 
11. }x—d3y = 38, 129, 
tea+hy=3d. ye —ty=4. 
13. Ly — 2x = 20, 14. iy— (=2%, 
La —2y=— 16. hie 4G) YH, 
15. te4+1y=8, 16.1%x+2%y= 


Lyte 10. 


146 SIMULTANEOUS EQUATIONS 


178. Elimination. —To cause one of the unknown quan- 
tities to disappear in the treatment of simultaneous equa- 
tions is called edimination of the quantity. 

For example, if z + y = 7, x — y = 3, then, by adding, 2 z = 10, 
and z=5. Here we have eliminated y. 

179. Methods of elimination. — Not only may an unknown 
quantity be eliminated by subtracting, as in §§ 176, 177, 
but it may be eliminated by adding, as in §178, above. 
We are at liberty to eliminate the ~ first if we wish, or 
the y first, as in § 178. If the signs of the term containing 
the letter to be eliminated are alike, we naturally sub- 
tract; if unlike, as in § 178, we add. 

It is sometimes better to eliminate by substituting at 
once, as in the case of a+ 38y=17, 8x%—y=1. 

Here we see from the first equation that x =17 —3y. Sub- 
stituting this in the second, we have 


or Dieta yee 
2. Hence —10y¥ =— 50, y = 5. 


3. Therefore z=17 —3y=17 —15 = 2. 


WRITTEN EXERCISE 


Solve the following : 


laa = 3 25; 2. 6x —y = 51, 
20 Oy = 11s Sa+y=47. 
3. e241 y = 39, 4. t= ype 19, 
5x+2y = 380. 5e@+3y=19. 
5. 28 yo, 6. Ta+4y=15, 
4e+ty=— 59. - 4x —Ty= 565. 
7. 2xe+3y=5S0, 8 «x«—sy=116, 


32— y= 20. 5a+ y=100. 


ELIMINATION 147 


180. Illustrative problem. — If to the first of two numbers 
I add 8 times the second, the sum is 21. If from the first 
number I subtract twice the second, the difference is 1. 
What are the numbers? 


1. If x = the first number and y = the second, 


Cee, 
eyes, 

2. Subtracting, LO — 20; 
Yeo. 
3. Substituting, eae ols 
mee ay, 


Check. Substituting in the first equation, because the second 
one has been used in finding the value of z, 


Der 16 = 21, 


9. Find two numbers whose sum is 99 and whose 
difference is 17. 


10. Find two numbers such that 7 times the first equals 


TA 


8 times the second, and such that their sum is 74. 


11. The sum of two numbers is 20, and if 14 be sub- 
tracted from the first, the result is the second. What are 
the numbers? 


12. What is that fraction which equals 2 when 5 is 
added to both terms, but equals } when 2 is subtracted 
from both terms? 

13. What is that fraction which equals 14 when 7 is 
added to both terms, but equals 50% when 6 is subtracted 
from both terms ? 

14. The sum of the faces of two promissory notes is 
$1000. The first draws 6% interest and the second draws 
5%. The sum of the interest on each for one year is $56. 
Required the face of each note. 


37. 


39. 


41. 


43. 


SIMULTANEOUS EQUATIONS 


tat ty =4, 
gu+ty =8. 
_Qa+38y = 22, 


3a+4y = 29. 


ee Oy Ens 
~Fe+2y =d0, 


Oe 2 f= 198: 


> hy, Ape 
SUT SEY =. 


je 1 — 
~4a+ Ly=8, 


se+yoy=l. 


3,¢ + y=101. 


 385a+14y=9, 


10 — ey a 
oe y— 1 
8a+16y=18. 
.842—Ty=19, 
3g —4y=19. 
txa— y=A4, 
fru —dy = 4. 
.18%—-107=9, 
27” + 40 y = 82. 
10 Syl, 
124%—5y=—4. 
ta+3y=Odl, 
2% —3y =-— 30. 
2+ 9y=Al, 


8x—lly=—1T. 


16. 
18. 
20. 
22. 
24. 
26. 
28. 
30. 
32. 
34. 
36. 
38. 
40. 


42. 


ye+ ty =4, 
sc+3y=3l, 
te — Ly 4, 
se+rgy=Y, 
40 +4Y=0 
TET ZYy=", 
$x — y = 18. 
tatiy=12, 
e+ 3y = 99. 
Ae+Ty = 22, 
AO t= ede 
3 « — 2 y = 36, 
ig Pa eLOr==eL 00 
Tx + 1ly=188. 
14% + 334 = 22. 
4xn+5y=4, 


12%—dy=—2. 


22+ 83y = 34, 
15a — 79 y = 255. 


424—60+3y=2e+ y+ 234, c=y. 


PROBLEMS 149 


181. Illustrative problem.—In preparing some medicine 
a druggist wishes to know how much water he must add 
to a quart of alcohol, which already contains 5% water, 
so that the mixture may contain 50% alcohol. 


1. Let « = the number of quarts of water to be added. 

2. Then 50%(1 + 2)= number of quarts of alcohol in the 
mixture, which must be the 95% of a quart with which he started, 
since no alcohol has been added. 


o> Then 50% (1 + 2) = 95%, 
l+a=%, 
£=33 -l==% 


4, Therefore he must add ;° of a quart of water. 


44. How many ounces of gold must be melted with 30 oz. 
of gold 16 carats fine (4¢ pure) to make an ingot 18 carats 
fine ? 


45. How many ounces of pure gold must be melted with 
18 oz. of pure gold and 6 oz. of silver to make an ingot 
22 carats fine ? 


46. How much water must be added to a quart of a solu- 
tion containing 8% acid and the rest water, so that the 
new mixture shall contain 6% acid? 


47. How many ounces of pure silver must be melted 
with 300 oz. of silver 800 fine (800 parts of pure silver in 
1000 parts of metal) to make a bar 850 fine? 


48. How many ounces of pure silver must be melted with 
40) oz. of silver and 100 oz. of tin to make a bar 900 fine? 


49. How much water must be added to a gallon of a 
solution containing 9% of a certain extract, so that the 
new mixture shall contain 4% of extract ? 

50. How much cotton-seed 011 must be added to a pint of 
a mixture, which is + cotton-seed oil and the rest olive oil, 
so that the new mixture shall contain { cotton-seed oil? 


150 SIMULTANEOUS EQUATIONS 


182. Literal equations. — Equations in which some or all 
of the coefficients of the unknown quantities are letters 
are called literal equations. 


For example, ax + by =c, mz + ny = p are literal equations. 
The letters a, b, c, m, n, p in these equations are supposed to rep- 
resent known quantities. 


183. Illustrative problems. —1. Solve the equation 


laa 


ax—Tb=e. 
1. Adding 7 6 to both members, 
OL = 97 Doe 
Tb+e 
x =—_——_-: 
a 


Le) 


(Check the result.) 


2. Solve the equations ax + by=c, be+cy= d. 
1. Multiplying the first by 6 and the second by a, 





abz + b’y = be, 

abx + acy = ad. 
2: (6? — ac)y = be — ad. 
be — ad 
ae y — = . 
b? — ac 


We may now find the value of « by substituting, or we may 
eliminate y instead of x. Taking the former plan, and sub- 
stituting in the first, 


b(be — ad 
4, ax alae!) =r 
2— ae 
b(be — 
b2 — ac 
b?c — ac? — b?c + abd 
6. ERAS A I ot A, 
b2 — ac 
a(bd — c? 
é _a(bd~ of), 
b2 — ae 
bd — c? 
8, e= ; 


b2 — ac 


LITERAL EQUATIONS 151 


WRITTEN EXERCISE 


Solve the equations in Hxs. 1-16: 








l. axe +b=c. 2. he aT 
3. abe + bex = abe. 4. px —GY= qe — p. 
5. axt+b=ce+d. 6. maz — nx = ax +- 0. 
1. mx + ne = px + 9g. 8) 2 me 4 NX — pr = YJ. 
9. «+ ax+ bxe=cx+d. 10. (a+b)ea+c=(b+c)x+d. 
ll. aw+y=m, 12. c+ay= 8, 
bx —y=n. ax+y =e. 
cy ea (ae ee 
ath ab 
xe+2 x—Y- 
ee saieps 
15. dax+3y=a4, 16. 4¢%+1y=2a, 
2x+7ay= 0d. oa —2Zy=0. 


17. One number is @ times another, and their sum is 2a. 
What are the numbers? 

18. The sum of a certain number and @ equals the sum of 
another number and 2. The first number plus the second 
equals c. Find the numbers. 

19. The sum of two numbers is m and the sum of the 
first and » times the second is ». Find the numbers. 
What are the numbers if »=3, m=2? 


20. If to a times a certain number there be added n times 
another number, the sum is 0; the first number minus c times 
the second equals a. What are the numbers? 

21. The sum of two numbers is s and their difference is 
d. What are the numbers? From the result write a rule 
for finding each of two numbers, given their sum and their 
difference. 


152 SIMULTANEOUS EQUATIONS 


Solve the equations in Exs. 22-40: 


Pe TU NS iene y ti 23. aa +7 ab = 02. 
24. abx — 8 abe = bec’. 25. aba + abe = be’. 
26. 4ax — 3bc = Qbe. 27. £2 2a — ao 
28) Ont — be 99. x? —4ab =a’? +467. 
30. 1627 —n?#=n7-+2mn. 31. 6 076% + babe=—A1 abe. 
32. ax +y=), 33. ax — by = 8, 

Diy —G bx — ay = 5. 
34. we — by =e, 35. abx — bey = ac, 

ax — by =. ace — aby = be. 

36. daz + 3by = Ge, 37. D008 2y ==, 

Sax + 2 by = Te. dace + by = 5. 


38. }e@+ar+aV=0?+4 az. 

39. 27 — 1247 = 360+ o- 

40. a? (a? — 6?)= ¢(2ab- ¢). 

41. If the sum of two numbers is 19, and their difference 
is 7, what are the numbers? 

42. If the sum of two numbers is 251, and their difference 
is 75, what are the numbers? 

43. If the sum of two numbers is 24, and one is 7 times 
the other, what are the numbers? 

44. If I have 15 cents more in one hand than in the 
other, and the total amount is 75 cents, how much have I 
in each hand ? 

45. Of two numbers, a times the first plus 6 times the 
second is k, and m times the first plus » times the second 
is 7. What are the numbers? 

46. Of two numbers, the sum of twice the first and 3 
times the second is 38, and the sum of 5 times the first 
and 6 times the second is 83. What are the numbers? 


QUADRATIC EQUATIONS 1538 


QUADRATIC EQUATIONS 


184. Solution of the complete or affected quadratic. — We 
have already learned (§ 168) how to solve the incomplete 
or pure quadratic equation. We shall now consider the 
solution of a complete quadratic equation like 

$9 -— 15 = 0. 

Peevacvorins go oho, we nave (e— 3)(7—-0) = 0. 

2. It is evident that this product cannot equal 0 unless one of 
its factors is 0, and that if either factor is 0 the product must be 0. 

3. Therefore the equation is true if 

x — 3 = 0, in which case x = 3, 
or if x — 5 = 0, in which case x = 5. 

Check. Substituting 3 for x in the equation, 9 — 24 +15=0. 

Substituting 5, 25 — 40+ 15=0. 


ORAL EXERCISE 


Solve the following equations: 

VY @—2)@—3)=0. 2. (« —1)(«—7)=0. 

3. (x — 8)(# —10)=0. 4. (x —9)(# —11)=0. 

5. («@ + 8)(@ —11)= 0. 6. («@ + 9)(@ —10)= 0. 

7. (© +7)(@+12)=—0. 8. (x + 6)(« + 20)= 0. 
Jee) =) 010% 7a a Orel ee == 0, 
125 (e+6)=—0. 13. eto On la eae a0.) 0. 
15. (2a —1)(a—1)=0. If 2%a—-1=0, 2u=1; then 

what does x equal? What is the other value of «? 

16. (32 —1)(# —2)=0. 17. (4a —1)(x —3)=0. 
18. (5% —2)(x — 3)=0. 19. (7x —3)(«#—5)=0. 
20. (8% —2)(4# + 6)=0. 21. 2e%—7)(3x%+ 2)=0. 
22. (2a +1)(8e%+4+2)=0. 28. (8x%+5)5u+4+ 8)=0. 
24. (2a —4)(38a%—9)=0. 25. (Tx —8)(8x—7)=0. 


154 QUADRATIC EQUATIONS 
WRITTEN EXERCISE 


Solve the equations in Hrs. 1-20: 


eg a 2 0) oe ee Or: 

BO? gh: 4. a7? —xa2 —12=0. 
5. 2? + bat Ss = 0, 6. 27+ 544+6=0. 
Vee Bs ee AI 8 a Oe, 
Sree ule ne al Bc OF 10. 27 +9272+8=0. 
Mea te 2, 12. 27 -— 92+ 20— 0. 
13.27 — 92 + 14 — 0, 14. 27+ $2 + 12= 0, 
15) 7 - 102 9 = 0. 162°97 4 2 
Lipa — 12%: -- 36'=-0. 18: a? = 112-4 30 =%. 
19. #7? + 12%+4+27=0. 20. +1274 35=0. 


21. What number is 16% of its own reciprocal ? 

22. Find a number which is 6 less than its square. Are 
there two such numbers? Are both positive? 

23. Find a number which when multiplied by 1 more 
than itself equals 12. What are their signs ? 

24. Find a number whose square increased by 35 is 12 - 
times the number itself? . What are their signs? 

25. If a certain number be subtracted from 16, and the 
difference be multiplied by the number, the product is 55. 
Required the number. 

26. A certain rectangle is 3 yd. longer than wide. If 
the width be decreased by 2 yd., and the length increased 
by 7 yd., the area is not changed. What are its dimensions? 

27. The width of a certain rectangle is 7 ft. less than its 
length. If the width be decreased by 4 ft., and the length 
be increased by 22 ft., the area is not changed. What are 
its dimensions ? 


ANSWER BOOK 


TO 


ALGEBRA FOR BEGINNERS 


BY 


DAVID EUGENE SMITH, Pu.D. 


PROFESSOR OF MATHEMATICS IN TEACHERS COLLEGE 
COLUMBIA UNIVERSITY, NEW YORK 


GINN & COMPANY 
BOSTON - NEW YORK - CHICAGO - LONDON 


Copyright, 1904, by David Eugene Smith 


ENTERED AT STATIONERS’ HALL 


COPYRIGHT, 1904, 1905 
By DAVID EUGENE SMITH 


ALL RIGHTS RESERVED 


35.10 


The Atheneum Press 


GINN & COMPANY. PRO- 
PRIETORS - BOSTON - U.S.A. 


ANSWERS 


TO 


SMITH’S GRAMMAR SCHOOL ALGEBRA 


Answers are given for the written exercises only. 


Page 5 


Leese eazoe. 63: 10, 4. 8. 5. 34. Ga S20 nlon Os 
8. 4. 9249. 107.840 11 212 12328. s18.188. 14, 711. 


Page 7 


Page 8 
Loe lo: 2. 4. 3. 15. 4. 14. Ory 6. 16. (he. VAS 


: Page 9 
Lael: 2. 12. 3. 16. 4. 15 5. 7. 6. 8. 
Sali 9.3: 10. 5. 1S) fe OE 12. 15. 

13. 4 14, 4, 15. 8 16. 53. 

Page 10 
Loo: e205 3. 25. 4. 3. 5. 9. 

Page ll 
LOU: 2. 200. 3. 75 lb. 4, 140. 5. $2400. 

6. $95, $9.50. 7. 2400, 360. 
Page 12 
“A 

eye 2eclds Sipal. 4115 5. 400. 6. 230. 
C213; S213, 9. 61. 1035 11. 3. L219; 


13. 33. 1492; LO uf; 1Giai Lien Loot 


OoOowPe 


GRAMMAR SCHOOL ALGEBRA 


Page 13 
. $72, $105. 2. $12, $50. Fg) Amit Fs 4. $90, $60. 
5. d = tm, d =140%; 6. rp, p+ rp. 
Page 14 
20-6 ea 3d 8135.) 4. 86 5. 125 —a 
127 + ¢. 1.4 8. 12. 9. 74 LOS. 
4] 12ao 13. 6 14. 55 15. 510°. 16. 11 
Page 15 
Sete 2. 17 ft. 3. $17. 4. 7b6—3a. § aie Geo 
pelts 8. 4 ft. 9. 363. 10, 357 sq. it. 11: 117514,000: 
. 6286 ft. 13°°27-5 in: 14. $236.09. 
15. 51,200,000. 16. $1.75. 
Page 17 
. ax lb., 2400 lb. 2. ary \lb., 108 lb. 
. abe dollars, $72. 4. Coefficients, 3, 4, 40%, 0.5, .652, 12.5. 
Page 19 
LOR LOS AMET 2% 115, 60;,0,.1 1: 


. For example, 8y=15, 3yz2=60. 4. For example, 4n¢g+8 m=100. 


Page 21 


me oUGs 2. 8a-+ 5b cents. 8. 6r +3747; 
.lw—r? 5. 5m + 82, 290; 826,865 sq. mi. 6. 57 108; 344 mi. 


Page 26 

ed Ose Lows 2. 173%, 1414. 8. 55 ab, 154272. 4. 128. 
Page 27 

. 67a4+ 440. 2. 150%+4+118y. 8. 155272 +172y. 


. da + 200 + 10c. §. 24a 4150 + 16c. 6. 9a+9b+ 9c. 
.4a+60+4+ Be, 13+ 23474+8=51. 

.§6a+8b4+ 382, 645410416 = 37. 

. 764+ 8c+4+ 72, 10 + 22 + 16 4 20 = 68. 


ANSWERS 3 


Pages 29, 30 
ee ee 2. 156. pe MiG 4. 64. 
5. 40. 6. 30. (ea G he 8. 87. 
a oy es 10. 9. De 12: 12. 8. ; 
1G Sen 14. 3. 15. 7: days. 16. 9 cents. 
17. 8 o’clock. 1S 28rd 10 1d? 19210: 20. 45. 
21. 97. 22. $4, $32. 238. $1380, $260, $1040. 
24. $1100, $2200, $3800. 25. $6, $12, $18. 26. 30, 10, 60, 60. 
Pages 32, 33 
1. 29ay —z. 2. 34m?n —11mn2 38. 144 pq — 16 82. 
4. 1038a2+5b2—2c%, 5. 0. 6. 20 ab. 
7. 223 a2zy?z + 14 p. 8. 4m?. 9. 6a? — 4 ab. 
10. ldary+4z2+w. 11. 20 pq + 20s. 12. 988a — 188b + 45c. 
18. 70 m2n + 168 mn? +110. 14. 99 a2b + 67 ab?. 
15. 48 rqr + 1539 grs + 491. 16. 29 abcd + 96 bede + 178 cdef. 
17. 100a + 100 ab. 18. 1004. 
19. 129 — 47q —176r. 20. —15%+ 63y 4+ 442. 
21. 155m+ 38n-+ 31 p. 22. 12% —182y +4 155z. 
28. 14932 + 1637 y — 149z. 24. 1900 a — 10006 + 1000c — 100d. 
25. 100a+ 100b — 50e. 26. 1052242 + 2938 ay + 1271 7. 
27. 5a+1864+4c, 12 +134 42 = 67. 
28. 43a2+ 662, 47 —17 + 67 = 97. 
29. 2a27+4b+ 4c, 24+124+16= 8380. 
Page 34 
1. 45% + 88 y. 2. 77 ax + 95 by. 
3. 5622 + 90 72. 4. 6.23 abe + 1.79. 
5. 24a + 426, 145 — 55 = 90. 6. 20a+12c, 1389 —68 = 76. 
Page 35 
Peelieeeee AS weeo wis. 4. be" 0,057) Goll ania ise) Seeds 
Pages 37, 38 
1. 8ay + 20yz + 7. 2. —4ab+ 2cd. 3. 34 m2 + 32 n2 + 10. 
4. 48 m?+ 35n?+ 2p?) 5. —Ta—8b+7c. 6. 40a%+ 706% 
7. 7a+564+2c+10. 8. 7Ta+170?—6c. 9. —8abe +11. 
10. — m2 — 222, 11, — 542+ 24y?. 12. p?+ 43 pg. 


GRAMMAR SCHOOL ALGEBRA 


. 1744. 14. 151 xyz. 15. — 26.43 4+ 128 a’. 

. — 162122 — 1. 17. 7 a2b + 6ab2. 18. 148a@ — 631 b. 

. 2m2+ 14 mnp. 20. 21.1%4+2.9y. 21. —69a4+2b—2c¢. 

. da2+ 11 a. 23. 77?4 2. - 24. 71 p?+ dq. 

.—- 24+ 6y—2. 26. —18%7+6y+4 8z. 

. 182a¢y + 42 yz + 56z2w — 81. - 28. 3423 — 4522-172 — 80. 

. 120a4 44b+4+ ¢. 30. —2a?+6ab+ 8b? 
Page 39 

. 1042+ 10 y?. 2. 3m’, 3. — 3abe + bed. 

. 8434+ lla —10. 5. 2m?4+2n2?+2a+y. 6. b—4c— 30d. 

. e+ ary + y?, 18 —(—1) = 19. 8. 10a2+ be+d. 


. 8—a@b+ab?—b% 10. 3a%4+ 2024 8c?. 11. 4% 4+ 2y? —2z. 
. 8224+ By% 4+ 22, 

. 6x2 + 11 y? — 2122, 129 + 344 + 395 — 125 — 498 — 159 = 86. 

. 21474 29xy + 16 y?, 74 — 258 + 1702 +1889 — 913 + 407= 2406. 


Page 40 
Ais 2. 2. 3. 18, 20, 20, 24. 4 Dt, Osta 


Page 41 


a?+ 2ab+ 62 — a?+ 2ab—b2=4ab. 

.m—2mn — 8n2— m+ 2mn + bn? = 2 n?. 

p+ g@— p?+q274+2 p?— g@— p?+38Q@= p?4+ 42. 

. — 6a°b + 6 ab? is the simplified result. 

. @b+b2e+c2a—a2b+ b2c —c?2a + a2b+ 2 b2e —3 c2a=a2b+ 4 b2c —8 c2a. 
522+ y? — 10 2? is the simplified result. 


Page 42 
. 2875 a. 2. 9683 a. Sue LOLry. 4. 1794 ary. 
. 240 abay. 6. 240 abzy. 7. 625 mnzy. 8. 625 mnxy. 
. abex + abcy, 5a+ 5D. 10. abx? + aby?, mna + mnb?. 
. ae+ bx, 2ax — be. 12. bp? + bg?, 5am + 5az. 
. apg + b3pq, ba + by. 14. 6 da. 9hr., 6x74 Dy. 
Page 43 
. a3d3, 2. a2b2c?2. 3. x22y + xy. 4. abbict, 
. on 6. 423 + 42y?. 7. 2m4*+ 6 mn’. 8. 60a’. 


. Baim’, 10. 16405 + 2464. 11. 2a%y + 2ay?, 12-5 — 60, 


ANSWERS 5) 


Page 44 
. 1227 — 6 xy. 2. —4a3+ 4ab?. 
21 x2y — 252 xy?. 4, 32 a*+ 32 a2be. 
— 15 atb — 15 a?b?. 6. — 41 a4 + 41 xy?. 
. —17a&m + 17 am. 8. — 45a2y — 105 zy?. 
.@t+tabt+actad+ae. 10. 756 a2be + 882 ab2c. 
. 26m3 + 389 m2n. 12. —atbc?d + a2b2c2d — a2bctd + a2bc2d?, 
Page 45 
. 15 pqs. 2. 9arxy?. 3. 16.4253. 4. 3q'. 
. Dax: 6. 3m. 7. 2men. 8. 24. 
. 24 bd. 10. 12 273y3z3. 11. 16 22y?, 256 + 4 = 64. 
. p= 800 rt + rt = 300. 138. 9=1897 +27. 
Page 46 
. C2 + y?, 2. 522+ Ty?. 3. 2a4+ 60D. 
. ne + py?. 5. c2 + a2, 6. a27+4a-+4 5B. 
. im+i1i. 8. 8a3 + 11063. 9. 1+ 2a 4.3 22. 
. 72 + 2 22, 11. 14 1382y. 12, 1+19y. 
.a@+3a2?+4a-+ 5. 14. 3a? + a? + 5a + 25. 


. wy2z + 2a2y + 8 yz? + 5 yz. 

. Saury2z? + 24 ayz + 11 w3y3z3 + 1. 

. dm+ 5m2n + 8mn?2 4+ 15 n?. 

. lett 17 a8y + 18 22y? + 23 ay3 + 87 yt. 


Page 47 
. 2a%e + 2 ax? + 2 ax’. 2. 4m2nt + 12 mn?2. 3. 4ay +2. 
. 2a2+ 2a. 5. 12 22y + 13 yz?. 6. 16 a2x?. 
. dd5axy. 8. 2m* +2 m? + 2. 9. 30 mnz. 
. 69 pq. Lie: 12. 6p2q + 6 pq?, 36-5 = 180. 138. 21 pq?. 
. 405b + 4a7bt = 144. 15. x4 + wy? + x22? = 14. 
42% + 224*7+324+5= 14. 17. a +a2?+a+1, 80, 15, 2. 
Page 48 
5 a?, 2. —d3zy. 3. —llm. 
5xyz. 5. — 1482. 6. 35q?. 
. —422 4 32. 8. —m+2n—3. = 9. 13 p?2 — 5q?. 
—-a7+32—-—9, Ble eee Bye ty, 12. —7m?+9m + 12. 


p—_ 


S 
rm or 6 O aT Cr HP CO rt 


oo Oo 


_ 
woneoa re 


GRAMMAR SCHOOL ALGEBRA 


Page 49 


. 923 + 199 22 + 19% + 35. 2. — 14 p2q + 298 pq? + 85 g? + 7 p®. 
. 4204 + 35122 + 17 2? — 80x24 159. 
. 1015% + 658 x3y — 86 ay? — 78 xy? + 98. 
. 227 + 8425 — 6275 + 162 x2* — 25023 + 316 22 — 288 2. 
. —daeoy + 9xdy? — 12 cty3 + 243 a3yt — 78 22y5 + Ol zy?. 
m> + 3m4n — 9m3n2 — 146 m2n3 + 19 mnt — 8 n>. 
. oat + 25 ay? — 13 ay? — ll yt 4+ 17 wy. 
11. LOST, Lies 1224) 6) oso 14. 12. 
18. 16. 55. 17a. 18. 14. 19. 408. 20. 148. 


Sa ae OVP earl e 23. $170. 24. 1380. 25. 2100. 26. 8200. 


Page 50 


~2-7-m-em-n-@-u-e-y, 7T-18>-M-M-M-M-M-M-M-N- NN: Z. 
~m(et+y), dx7(8e+ 7y). 4. pq(p+q), 17 mt (8m? + 22). 
. dp'g(p+5q?+ 79°). 6. ary + y2?+2+a + a). 


Page 51 


. p(it+tr), $200 (1 + 0.10) = $220. 
~8=ce+cer=c(1+7r) = $1756 (1 + 0.20) = $210. 


Page 52 


. “(2 —8). 2 mi(m—n). 38. at(a+2b). 4. a*b(abr + y). 
. a(1+ b+ 0). 6. pq?r (p?q — 1). 7. e(v? +2247). 

. p2(p?+3pq4t1). 9. 24(402?+2241), 10. x(x?4+ 15% +16).. 
. 3m2n2z (m2n + 22). 12. 3y(9xy? + 3x7y + 1). 

. abc (ab? + ab + c). 14. 4m?(4m+38n-+ 2). 

. 5cd(ab + 2ef + 8cd). 16. 17a(a+ 3be + 9a?). 

. Tpg(sp+5qt8r). 18. x(78 + 842+ 4% +4 121). 


Page 53 


. abcauy. 2. ab, 3. pquy. 4. pqrs. 

. mnpxy, 6. a2b2c?, 7. x(a-+ b). 8. pgq(p+q). 
. abc (x + y). 10. 27 ab (a3 — 8). 11. ab(x — y). 

. cy(m+n). 13, 123 ab (a? + 262). 14. mnp(m + n). 
15. am(m + bn). 16. mx*(x + 3y). 


Li: 


14. 


. 18abc(8a+ 5c). 

. l7uv(u— 7). 

. (a2 + 8ay + by?). 
. a(a? + 8ab — 40"), 8. 
. d(abe + bce + efg). 

. 27mn (a + 4 pq). 

. mnp(m +n — p). 

. D(p — 8p? —.4 p3 + 5). 


ANSWERS 


Page 5 


. — par (pq? + qr? + 1). 18 
. 2Lab(2a—38b4 8c). 20 
. of pg(p + br — 2). 22 
. For example, 5 p2qr. 24. 


. For example, ma —mb+mec. 382 


. 47d. 39. 15 ab (a? — b?). 
. 27p? (p+ 2). 42. abcde. 
. d2abe(2a+b). 45. 60 abcd. 











4 


. mn2p (m2 + p?). 
. r(pg — gs + St). 


. Stay (dry + 1). 


43 x2y(22+5y). 

. “(v2 — day +d3y?). 

. 19m2n2 (8m + dn). 

. m2(m2 + n+ 4mn?). 
. 22(8e+4y+ 52%). 
. 41 p2q’r (2 p — 87°’). 


. 84 (422 + Oxy? — 16). 


. — 2Qxyz(4a2y + 3 xy? + 2). 
For example, 3 aq?rz. 

. For example, 12 a262c?. 

40. 25ab(a + 0). 
43. 60 p?q?r. 

46. abed(a+b+ 0). 





. Lbp?¢r(p+qt+r). 48. 3a?(a? + 2ab+ b?). 
49. 2a(a? — 3ab + 0?). 
Page 56 
; 2 2, 
a py an Pees Anes pe 
b y? 45 a? 5 avy 
2 2 2 2 5 

é 4x v+y gy eo 9 2 + qs 10. 2? 

9 atz3 3a a ps 4 

2 1 22 
8m +5n. eee ee cee Sane 
fies b 
3 2 2 3 ) 3 2 9/3 
Paty ot Pde rae Tae eee ee 16: See + 12 ry + 23 y3 
r 3 
Page 57 
2 — 
maeige el. Seidler eine) cere 
4 2 

b c 
. —- — —— dollars, $1.25. 4. 10,id,r=d-+e. 

ao. 100 
. 80 Ib. 6. 50 Ib., 25 lb., 100 1b. (AREER ; 


“m(p—qtr) m(p—q+r) 
. Numerators, ab (a? + b?), a? (a? — b?), b? (a? — b?). 

. Numerators, b?(a? — b?), ab (a? + b?), a? (a? + b?). 

. Numerators, be(a+b+c), ac(a—b+c), Pb(a—b+¢). 
. Numerators, 6 abs, 3 bep, 2 cdr. 

. Numerators, ab, b?(a — b—c), ac(a —b—C¢). 

. Numerators, 2 abcx, 2cy(a + b), abz(a + DB). 


GRAMMAR SCHOOL ALGEBRA 


























Page 59 
2 ab? 2. Le 3. nenp? 4. 2qrt 
4 b3 12 g2r2 m2n3pt 2 pg2r? 
ab ade 2 pq aes axy + bay 8. x2y2z2 — yar 
b2 + be 2q? —2qr axy — bry v2y2z2 + y2z2 
ab?m 382 11. ane Les be? 
ab2m? — ab?n? 3 w? +322 — 6we abe abe abe 
an bm 2% a) ea per? pq 1 
"2bn’ 2bn 2bn a a8 * p2q2r2? p2q2r2? p2q2r2- 


pqr PAD tT ate 


“araqt” ar(atr) art” 


LYZ xz (y2+ 27) xy (y? + 2) 


"yz (y2 + 22)° yz(y2+ 2) yz(y2+ 2) 


am AC ie ac 18. Numerators, st, 2 pt, 3 pq. 











Page 60 
Only the numerators are given to Exs. 1-6. 
1. rq+p. 2. 8m2k + n?2. 3. 2a°b? ++ be +a. 
4. ab —8ab?+b3. 5. m2n4+3mn242n3. 6. xy 4+ 2ay?2+ 34+ w. 
" 6 p?q + p? 8 12 m2nz + mn 
Thee , 3x 
9 4 ad? + 2bd?+ ¢ 10 16.x2y + ldxy? + 3x 
7 : aia ae 
2 Aree SWeinyes Sk ese 3 3 
11. 32a2y — 1l5y +t 12. 185 a 75 ab + 166 
y oa 
Page 61 
1 Lad 
1. 227+ 5%+4+-.- Ee 3. m+2+ ae 
o 16m 
3 17 b 
4.2@+34 


’ 5. daeacoeees 6. 2a+0+ 
x 


2 a2 6 a? 


Only the numerators are given to Exs. 1-9. 


IP eR 


13. 


ON. 
. 126 m2n. 
. 14a + 280. 


10. 


1 


4 py 
5¢ 








ZC 


ay +2 ay? + 4 


y 


eel. 


6. 22. 


_ 


60. 


ad + be 
bd 


3 a?beq + 2 b?cdp 
6 pq? 


. 308. 


7 00; 
. 380. 


5. 


ANSWERS 


Page 62 








-on + 9+ 


. 10 pq. 
. 4 q?rxiy?. 
. 2xyz + y2z. 


n 
2 

w+ ie 
m2 

38m + on 








OD: 


= 
o 

9 
eS ea 
Ov 


25 
On 
ad +bd+e 
; ad 
82 xy? 4+ 2laryi+ ex 
y 





Pal 5. 24. 
OU: 
231. 5. 300. 


abew + abcz 
WLYZ 


6 a? — ab? + 16 b2c 


24 xy. 3 
. tarbcyz. 6 
. 52 (p? —4q). 9 
ga” 12. 
7 be 
ees 15. 
3p+2 
bat by 18. 
i 
. 9a. 21 
. 9x+ ae 24 
9 xy 
2 
. Ay + ae 27 
8 xy 
4 
5pt+ 4g 30. 
5 p2 
6 5 
212° +172 a 33. 
2 
Page 63 
3. 35. 4 
S200: 9 
Page 64 
3. 400. 4. 
Page 65 
m2 + n2 
mn 
9 a2bn? +16 a2b2m 
12 am?n2 
38a 8 47 x3y? 





2¢ " 36mn 


4 are 


10 


26. 


27. 


GRAMMAR SCHOOL ALGEBRA 


ad — be 

bd 
10 abd — 9 abc 
12 202 





o2 =f y? 
LYZ 
mng — mn2p 
pq? 
ay? + bz? a+b 
ay? ; 


ile ofa lard ae 


par 


2e7y + 2y3 — x + ry? 


4 xy 





acdef — b?def + bc2ef — bed?f + bede? 


Bayz + yz + wz? + xy 


LYZ 


4. 


ez — az + ytxe — yx + zty — zy 


LYZ 


Page 66 . 
9 a? — c? 3 2ay — 2 bx 
"abe Zay 
5 1 6 2mxz—8nxe+ 8ny 
Cite ; 6 m2n2 
Page 67 
2 — 2 2 2 
gm Da g, Hod + abrc 
mpg c2d? 
5 a2 + 2ab — b? 6 azx + ba — x 
} 3 a-b? aby 
8 6mn + 5 n? — m2 9 ab+b+a—-—a 
35 mn a2b2¢ 
Dares, 
EE Pee thee 
q abed 
15. bd + cd — ab—ac 
abcd 
bcdef 
18 2ab +-2be + 2ca 
abe 
ab? + 6? —ab+b+a—l1 
20. cae oc een 
99 a—@+a}—b+b?—68 
; ab 
PY gaan 


pa er Ls rer ey 


pqr 





ac 


abcxyz + bedwyz + cdeuzv + defurvw 


UVWLYZ 
2 a2c? + 2 bc? + 2 a*b? 





arb? 


az2w2 + bz2w? + cz2w? + dz2w? + bxr2w? + ca2w2 + dxr2w?2 

















wx? y2z2 
+ C%2 w2 + ca2y? 4 dx2y? +. ex2y/? + fxry? 
wrarry2z2 ; 
Page 68 
asb3 x+y a—bd ab — a? 
" ¢3q3  ‘Tmay "16 ney edn. 


21. 


24. 


16. 


19. 


. $. 








ANSWERS ala 
Page 69 
ac? ac 3 (b+ c) a3b3c3 b azn? 
b2d bd? "mn  mnsx3 zg c2z2 
Page 70 
Be 2.9500. 3.97. 4. 14 mi. heey 
b bh 2¢ 
Page 71 
3 
bed 9, ade. eae hy Wace pete 
as bcf ns?qr? yz 2 
= Shyns 2b4e ane 
abmn (a 0) ow. 9 a’bp?q” 8. _ &btc aces 19, nt 
3 10cr det qg atbtc? 
12. 1. Wap e » § Cy) aa a 
de w mn? 
2b (q2— 
a*b (a®— b) 7, 1. ; deat 19 eee 50) ss. 
mn? LY a2b3c3d3 6 ptgtrt abary> 
8d (q2—b?2 
LAC sala Pp > nap esl a a 23. (a — y)2xyz. 
37 8 
abcxyz (a2 + b? + c?) 
pqr 
Pages 72, 73 
893 2 
ye 9, PT, Gopeen 
a mn? dx? 
ab(a +b) 5, m2 (p— @) rae 
cd? pqry ar 
2m (p? — 4q) P m3 (m + n) 9 2¢(a — 2b) 
py ; ; ays ; a2b2 
4(q2 4 p2 2 
CALs salle Ties {apo ee 
més yn a2b2 
4 2 
bd gy, Ae 1500. 
ac m? c2d? 
2 3 
pat 17. ners 18. x2y2z2. 
pq? abc 
Se oA 292 802 1 G2y8 
resent 920. pres* OS aid ae 21. a2e + ab?. 


Pq 

















3 











2 
22. —(l+a+ a). 23. a+b+c. 24, Bei 
abmn 
25. 2(xz — y?). 26. n2— m2. Q7; ite! sree 
a2h2c2d2 
28. pr+ar+pa+pr. 29. afd +bde — cbf. 30, “4. 
urv2w? 
81. (x — y)2. oe apy ke, gat 
ab2c?d7e hd wine 
Page 75 
1. 42, 2. 10 3. 6 4. 1200. 5. 800. Geite 
eel; 8. 12. 936: 10. $1200. 11. $800. 
Pages 76-78 

1. 2. Pe yep See 4. 12. Sa: 

6. 9. 7. 42. 8. 108. 9. 168. 10. 23%. 
loro: 12. 310. 13. 210. 142712. 15. 42. 
16. 28385. 17. 70. 18. 7. 19. 90.8. 20. 5. 

21. 550. 22. 13. 23. 15. 24. 3. 25. 1. 
26. 90. Pi hee hb 28. 15. 29. 50. 30. 87200 
31. $7500. 32. $7100. 33. 56. 34. $2560. 35. 200. 
36. $2480. 37. 900. 38. $950. $9.57; 40. $1750 
41. $350. 42. 21. 43. $2550. 44. $2400. 

Page 80 

1. a2+2ab + b?. 2. 274 2ay + y?. 3. m2+2mn2-+ ni. 

4. 4a? +4ab+4 bd? 5. a? —2ab + v2. 6. x? —2ay + y?. 

7 9m?—Gmn+n% 8. 9 —122?+ 1624. 9. a? — 02. 

10. a? — 02. 11. vw -—4y?. 12. mt — 9n?. 

13. 2a? — ab — 602. 14. 14 a4y4 — 19 x2y? — 3. 

15. 3024 — 1322 — 3. 16. 12 m2n? + 19 mnzy 4+ 4 2x2y2. 
17. a2 4+ 2ab+4+ b2 + ac + be. 18. 224+ 822y + Say? + y?. 
19. m3? — 3m2n + 38 mn? — nn’. 20. 8a? + 12a2b + 6 ab? + b. 
21. 621, a? — b2, 27-23 = 252 — 22 — 625 — 4. 

22. 1764 sq. ft., f2 + 2ft + t?, 4024+ 2-40.2 + 22? = 1764. 

23. $729, # + 2ts + 8%. 24. 96, etc, 


GRAMMAR SCHOOL ALGEBRA 


ANSWERS 13 


Page 81 
Only typical answers are given for pages 81-84. 
1. p? +2 pq + q. 4. vt — 227t + PP. 9. min? + 2m2n +1. 
12. ptg?r? + 4 p2qr + 4. 13. 4a? + 12ab 4 96. 
14. 9a? — 12 ab + 4572. 15. 252?y? + 10 ay + 1. 
Page 82 
1. ab? + 2abe+c?, 2 9—6yF 4+ y® 3. y+ 2y> +1. 
4, 4—475 + g1, 5. a2+10ac+25c2, 14. xty® + 27273 + 1. 
18. 9+ 302% + 2522, 21. wey? + 2 wryz + 2. 
24. 9m4 + 380m? 4+ 25. 
Page 83 
3. at—1 5. a2b2c? — 4, 7. 1 — mxt. 9. 9 — 1626. 
15. 144 —-49 = 95= 19.5. 27. (m* — 1)? = m§ — 2m4 + 1. 
Page 84 
9. pig? + 17 p2q + 70. 12. x2y2z2 4+ 17 xyz + 30. 
Pages 85, 86 
1. 2322 — 370% — 357. 2. 1892? + 39a — 518. 
3. zt + 2922y + 210 y?. 4. 25524 — x2 — 56. 
5. 25522 — 72ay — 12 y?. 6. 52522 + 100 cy — 161 7. 
7. 22 y2 — 125 ay + 22 22, 8. 527 x2y2z2 — 1838 xyz — 14. 
9. 1385 abbtc? — 57 a*b2c — 56. 10. 810 xty* — 51 x?2y2 — 3. 
11. — 18122? + 183 zy + 24 y?. 12. 75999 a? + 7707 a — 25872. 
18. 48471 at + 962 a2b — 85 b?. 14. a + b?. 
15. a3 — Bb. 16. a2 4+ 8a%b 4+ 3 ab? + BD. 
1%. a® — 3.a2b + 3.ab? — b3. 18. at — bt. 
19. 7871 a2b2c2 + 1286 abc — 17. 20. 4836 a2b2c2 + 264 abe + 3. 
21. 823 — 2622 + 25% — 6. 22. 322° —-16 242 + 142 — 3. 
23. 2142 a? — 2601 ab + 629 B?. 24. 10767 atx? + 2370 atx + 63. 
25. 823 — 12 a2y + 6 xy? — 7. 26. 2723 + 27 xy + Day? + ¥3. 
27. a + 12 a2b + 47 ab? + 60D. 28. 29584 xty* — 24, 
29. 714 ptqtrt — 25 p2q?r2s2 — st. 380. at + 403d + 6 a2b?2 + 4 ab3 + F. 
31. 6323 + 46a2y — 34ay? —12y3. 32. a3b3c3 — 3 a2b2c2d + 3 abed? — d?. 
33. a2x6 + arhaty? + ax2hyt + axrtby? + abx2y3 + by, 


Se 
woonrt >» = 


wo Ot Pp KF 


bob 


GRAMMAR SCHOOL ALGEBRA 


. at + 40° + 6 a2b? + 4.ab3 + DF, 

. 29063 + 8 aSbtc? + 3 a8b?c + 1. 

. 4803 — 74 x2y + 421 zy? — 150 y4. 

. wt —4axiy + Oxy? —4ay3 +4 yf. 

. 8192 x6 + 1381 x* — 934 22 + 41. 

. 567 xtyt + 48 w2y2z2 4+ 28, 

. 1829 — 83 a5y — 89 a8y? — 56 y3. 

. 216 a8 + 756 a2b + 882 ab? + 343 b3. 


Page 87 


.6. 2.4. 3.3. 4.9. 5. $950. 6. $250, $225, $375, $562.50. 
- 34%, 13%, 42%. 8. 6%, 4%, 3%. 9.938 ¥ru, 2.2 Yolen 2 eey te 

. 22%, 14%, 83%. 11. $3850, $346.815%, $340.62, $366.82%. 

12. 4%, $270.40. 

















Page 88 
- (a+ Dd). 2. m(a-+ Db). 3. v(v+2y). 
- a(p—7). 5. xy (x + 9). 6. p(p— 34). 
. bc(a+ a). 8. ax(x~+y). 9. b(a+ 5c). 
. 8q?(p+24q). 11. x(mx — ny). 12. c?(c2+ 3d). 
. 2y (a2 + 3 yz). 14, xy (ab + wz). 15. c(my + nz+ qu). 
16. 4ay(8y +22 +4 2?). 
Page 89 
mad g ttl. pees At eee 
m—1 x—1 a+1 y+2 
pls eas ets es 
n y+4 Yr Yy 
eA) oe Sse eee 19, Lee 
7T+2y z 3spa+tr 
Page 90 
. (2 — m)?. 2. (x + n)?. 8. (7? + 1)2. 
. (1 — 2?)2, 5. (2¢% + 1)? 6. (1 — 22?)2 
le eee ye: 8. (4a — 1)?. 9. (xy + z)?. 
(ab — cd)?. 11. (pq + 10)?. 12. (abe + 11)?. 


(5 a? + 7 b?)2, 14. (622+ 5 y?)2. 


ANSWERS 15 


Page 91 
- (p+ 2q)(p — 29). 2. (8a+2b) (8a — 2d). 
~ (4274+ y) (427 — y). 4. (8m?n + 1) (8m2n — 1). 
. (7+ 11m) (7 —11m). 6. (1 +10 abc) (1 — 10 abe). 
-(a+b+c)(a+b—c). 8. (x + a)2. 
. (6a + 5b) (6a — 5D). 10. (12m + 5n) (12m — 5n). 
. (6 pg? + 11) (6 pq? — 11). 12. (9 + 5 abcd) (9 — 5 abcd). 
. (a? + b)?. 14. 25b?(a +c) (a—c). 
- £(a2 + b+ cy). 16. (2m? + n)?. 


. (a2 + 9 cd2) (8 a2b — 9 cd?). 
. (a—2b)2— 42 = (a — 26 + 4) (a — 20 — 4). 


. (a+b+ 2) (a4 b — 2). 20. [1+38(e+y)P=(14+82+3y)% 
Page 92 

. (x + 7) (x — 6). 2. (v + 2) (x + 5S). 3. (% — 2) (x — 8). 

. (a— 9) (a+ 2). 5. (p+ 9)? 6. (a7 + 9) (x? — 2). 

. (m+ 7)(m + 5). 8. (p — 10)2. 9. (pq + 4) (pg + 9). 

. (ab + 10) (ab — 4). 11. (xyz + 3) (xyz + 8). 

. (mn + 9) (mn + 8). 13. m (n* — p? — q?). 


.(a+b+c+d)(a+b—c—d). 15. a?b° + 138 abscd? — 68 cds. 





























Page 93 
a+b ry mee 3. 1 re elem 
ab a—9d v?+1 a+2 

x uss 4 
ile ais pe eee aoe yea 
x+1 p—6 y* 

2 — a aan 
ar 10. 4% ye 11. 1 Ta 12. 2m n 
c2 xz — 20 1—9z2 2m+n 

Page 94 

mol: 2. $763. 3. $768. 4. $9750. 

. $10,750. 6. $8200. 12.3%, 8. $5500. 

. $90 each. 10. $80 each. 11. $9000. 12. $27,500. 
Page 96 

oY. 2. 2m—38n. 3. 0+ y. 4. 2p—4q. 


. 6r-y,. 6. a24+2ab4+ 0% 7. 5a—2b. 8. 20pqr+1. 


. 4ey+7 
. 1—Tpgqr. 


a 
4 
5 

.m+2m— 5+ 
m + 
+ 2ay + y? — 
. da+2b+4 — 


. SO) SS = 


- 4m+n— 


. 8 — xy + ey? — y?. 


GRAMMAR SCHOOL ALGEBRA 


10. 
14. 
Lit, 


2m? — 15. 
2 x2y? + 8 22. 
w+ y. 


11. 38 —17ay. 
15, 222 — 
4% —38, 4027+ 527 — 6. 


12. 2ay + 32. 
38, 8x3 —2a%7 —127+43. 
18. 38x —y, $2. 


19. 3872+ 4ay 4+ y?, r+ y. 


Page 97 


2 10 7/2 
oy »x—-syt+ ee. 
w+ Y 














Ty? 
+ 





oe cea Daca spr 





6p —q 


i TM TS 5 
y 


10 ab + - 


2 CUM nas 


oe 





~y 
oh es + 5b 


2a+b 


3 b2 20 ab + 5b2 
iD 
38 pq + 7¢ 


ox say —13y? 


3mn i: ae n2 
2m+7n 
n2 
seein SAI, 
5m+2n ‘ cs 


Page 98 


a —xy + y?. 2. 
4. 
= xy + xy? ea xy? + yt. 6. 


.@+2a4+3. 8. 
. 16mt*— 
. (&+y) (4 — y) (2? + 9’). 
aA2p +41) (29,1) (4 p24). 
. (3a +2y) (82% —2y) (9224 4y%). 

. (cy + mn) (cy — mn) (x?y? + m2n?). 
. Bpgr + 1) (B par — 1) (9 p?q?r? + 1). 


10. 
12. 


8m3 + 4m2—-2m+1. 


3xr+2y 
3mn — 12 n? 
2m +5n— 
21mn + 3 n? 
Bm—2n | 


at + asb + a2b2 + abs + dA, 
4¢7 +2241. 

1-—3a+4+ 9a. 

1+ 4ay + 162272. 

a?b2c2 + abed + d?. 

(2m + n) (2m — n) (4m? + n?*). 


ANSWERS 



































ae 
Page 99 
Only the numerators are given. 
1. 2 —52—14. 2. 2 —52+6. 
3. 28 + 622. So (0 — See). 
5. 6a3 — 5a? — 22a +4 24. 6. 8a? — 8la?+4+ 78a — 56. 
7. 6a —19a?+41la+6. 8. — (8a’ — 7a? — 2a + 8). 
Pages 100, 101 
Oe we D) 6 
TE a et pat ee pe ee VD, 
4 i) m ye 
6 2x3 + ay " ary +b 8 am + an + m 
Oy gy m+n 
eye. 2 ta, Prtueeta oe 
9. Y ae, 10. m Aas 11. m mk 
y m m 
ENG 72 72 Ieee ie bee ce 4 sae 
19. 2b? 13. a2 + ab = ab —b Cc 14. a+oed+a le 
a—b a—b cl 
3 g- evn? Sek) BE. pote Ot La 
15 c 16. az+a 2 17, Ut# 3H + 2 
x+1 a—1l Ag 
9 2 © 2 6 Dee 
fig, aay oe ee 19, -@ —8ats 
Dg a—2 
2 nt — 33 Dina 6 ot ipl ee 
20. 3p 3p? + 21p 8 21. e4+at*+ 3a 18 
3 p? 2 
4 3 2 = 2 3 Diy eee oie Te. 
go Bteta+a-1 Ps alle ova pL ey 
a Pg 
2 46 — 245 BRAC Sn 2 x8 2 2 
94. oo 3a°+ 3a 2a - 25. 22 + ay + ty? 
az+1 e+ y 
wer} ve 2 ie — gq2y72 — 2b 
26. 6m + 17m O17. 124 +22 21 28. arn 2b 
2m 3 4x+7 ap 0 
— 2m2 2 € Dy o 73 2 2 2 
29. 2m +n? 30. SUA c 31. 2a2+ac?+a ale 
m—n 2ny a+l 
2 Qp2 _ 33 ‘ 
39. 2 a2b — Gab 33. 10 a2b? — 14 a2b + 8ab+2 
a—b 1— ab 
34 16 m> — 7 m4 — 15 m3 + 7m? — 2 35 8 x3y3 + 5 aty? —4ey4+1 
m2 — 1 , ry+1 
36 m* + m3 + m2 —m + min + m?n + mn + n 





m+n 


18 


37. 
39. 
41. 


43, 


" 2 (2 p 


13. 


GRAMMAR SCHOOL 


2234+ 38x2y + 3827? 
ty 

SDA gd 

prd 

— 5n? 
3m—5n 
3552 a2 — 777 b2 + 666 c? + 37 

111 





Page 102 


a OU ae cere 2. a 








» pg+8. 
. a2+ ab + b2, 
a2 — ab + 02, 


Page 103 


a+od a? 


2pr Pp") 
—r) r(2p—7) 
a(a?+ 1) az#—1 


" @2(a2— 1)’ a2(a2—1) 


a(a +c) b? 


“be(a+e) be(atc) 


x(x + y) oe 


yz(ety) yz(ety) 
Li 


"(1 100). pay 
pg(l— pq) p*q2(1— pq) 

yore ete 

wy (x+y) cy(@@ty) 





15. 22. 


38. 


40. 


42. 


44. 


ALGEBRA 


3273 + 4007+ 42% —5 

e+1 

2ab — 6? + 8ac4+ 3be 

a+b , 
—4ab+0?+2ac 
a—b 

28 a2 — 20b? + 10 c? — 

3 


20 d? 


2 b4 


3 — q2b + ab? — b? + ——__.- 


a+b 








11. zy + 4. 
14, 7?4+22y 4 y?. 


m+1 m2 

3 m (m + ey m (m+ 1) 
a(y + 1) bx 
xy(y +1) ay(y+ 1) 
m2(m?+1) m+1 
* “m3(m +1)" m3(m+1) 
m(m + n) n 
n2(m + n) "7 (m + n) 
DY Ae Dae 
p(p—4) p(p—4) 
BAe Dh ie: 
@r(l+p) Pr(+p)’ 
(m— 1)? m2 


m(m — iby m(m = 1) 


12. 








14, - 





ANSWERS 13 






































Page 104 
e+7 9 ¥t4 3 p+2 mare 
“g+9 yore Ges Bon 2 
%— 4 6. pq —7 7. cy —4 g ttBy 
x—6 pg —9 zy —7 c+t4y 
mn — 21 10. abe — 2 fhe Aas 12. mn +7 
mn — 22 abc — 5 pg —2r mn + 8 
Page 105 
v§4+273+222? +2741 9 —~e+at4+r3—g72?—24r4+1 
e+] : 1 > 
m2 — n? — pm + pn — p 4 men? — 138 mn + 2 m2n — 26+ mn? 
m—n mn + 2 
pg—1 c—y 
99 x8y7 — 158 xoy* — 17 + x? 
Day 
528 ml} — 391 m4n}2 + 391 m1 — 288 nl? 
23 m* +17 
x + 23 — 3 y—1 
. £8 + 43 + —______.. 10, y3 — 3 : 
ep Sey ESE Re Le aera 
6 aod 37/3 —_ 
Re es Ce) oe ane) eee 
g-+11 wy +1 
9 y* 13 
. 8+ 327%y + zy? — xy? + ——. 14. por —16— 
a+ y aoe 
eee nee a are 7 817, Ce 
mn — 5 w(e+y) x(x+ y) 
a(a—1) a—-1 VO et Py eee) s 
a a ay(e—y) ay(e@—y) 








2ey(e+3y) («+ 2y) (4a — y) 21 ab (ab+1) —(ab—1)? 


“6ay(v+2y) Gay(x +2y) "a3 (ab—1)' a'b3 (ab—1) 


BAP Gc PG) (Pp — 2944), 
4(p — 9q) Ago) 


Bap — Apr dg?) (4p +9) (pb? gs) 


2G(4p+q) 2q3(4p +9) 
a(a+b)? (a —b)8 
ax(a — bv)’ ax (a — b) 


20 


22. 


GRAMMAR SCHOOL ALGEBRA 





Pages 106, 107 





. 4, 














2e+y a2 + 2ab — b? 2m+n 
ox (e + 9) a(a — b) " m(m +n) 
222—1 5 2 6 m+mn+n 
 @ (x? — 1) ob "-n(m+n) 
ere Pele Py gee eat EN 9. 21. 
b(a + b) p(p+q) 
— 292 22 
_ $1200. 11 6pg— 8g 12. Oe Die 
p*(p — q) abc (b + c) 
am+4a2-4+ 22 14 acde + b?de + bc?e + bed? 
"m2 (m + 4) bede 
v3 + 72 — y? 16 e2+eytatyt at — xy 
a? (x + y) . a? (x + 1) 
2 (m2 + mn + n?) 18 222y — 2Qry? + 3 
"m2 (m + 2n) xt (x — y)? 
8p —q4+8p24+11pq4+6¢q? 20 5 3 + 29 7? + 98¢ 4 11 
3p (3p + 29) x? (aw + 2)? 

180. 22. 282. 23. 3. 24. 9. 2OpmLte 26. 52. 
veld 28. 4. 29. 7. 30. 5. 31. 48. 32. 5. 
33. 12. 34. 10. 35. 32. 36. $1250. 

Pages 108, 109 
patel Ye Ee ett al g Pte —py-1 

x (% + y) a (% — 1) q(p — 1) 
abe — axy — aby + bry 5 pg —3p—3q 

by (b — y) pq (p + 9) 
a3 + a2b2 — 2 ab — b? a2 + 62 —1 8 —2b 

b (a2 — b) abe "(a +b)? 

2 Roos 

cit lara 10, @ 2 2@t!, Oh. ih 
P(p — 4) a(a + 1) 

138. 10,192. 14. 2. 15. $250. 
ay" + yi — xy — 2% ay euciey 1g, PY PT PT 
yy + 2) a (a? — 1) Oped) 
pg — 8p — 21 20 ayy 21 2y + vy — 227 
" -p?(p +7) at (a + y) y (x2 — y) 
x+22-—624+9 3 axy? — by — bz 24 3 (a + 2) 
22 (2% — 3) "  g2y2(y + 2) " a2 (a + 8) 


25. 
28. 


13. 
19. 


ap ALTO ONE 




















a—2b 


ANSWERS a | 
vty — 2+ y3 26 3ab+3 6? — a? a2m — 2 b?m + 4b 
xy? (43 — y?) ata 2p) 6 m2 (m — 2) 
30. 29. 30. 30. 14. 31. 21. 32. 70. 33. 33. 

n Ad. 35. 56. 36. 200. 37. 35. 38. 625. 39. 28. 
Page 110 
e+ 9 Pa(e+y), 3. (a +d)? Den ae 
c—y 15z n(2m + 5) 
5. eae 6. (8mn — 4)". 
Page 111 
a(a — b) v2+1 q x 
b(a + db) oes tl 6 p y (x — 1) 
Bip hee bts 
3(a + 6) b> 
Page 112 
28. 2. 160. 3. 33. 4. 28. 5. 54, 
210. 7. 40. 8. 4. 9. 36. 
Page 114 
8.4. 2. 37.5. 3. 15. 4, 5.5. 5. 1. 6. 14. 
1s 8. 0.02. 9. 50. 10. 80. 11. 33);. 12. $1000. 
320. 14. 20 15, $1200. 16. 32. 17. 5. LSS: 
2 20. 51 21. 60. 
Page 115 
ahi, a1: 0 ee 4. 8. 52 6. 6 
Page 116 
2 = 
88 213 ; bal) Aer: 5. Gat + 6ab — 2a 
a+ob+1 3a+b 
bs ‘ 3 a2 
SSO Dy tana Al) Cys iy 


AGP gt =sh3 
ligne 2716; gp) OO O81 glee Ce ca eee 
d+ ac a+ob-—1 
6. 15. 7. 50, 40. 8. 75 yd. 9. 60in., 40 in. 10. $60. 
11. 30, 20. 12. 4. 13. 10. 14. 24. 15. $1100. 
16. 10 rd., 29 rd., 290 sq. rd. 17. $1500, $8500, $5000. 
18.2. 74ns, 3.0m. ee ip: 19. 450 mi. 20. $5000. 
Page 120 
1g? re NY Sali. 4. 30. 5. 1383. Cin0, 
Page 121 
1. 8 ft. from 56 1b. 2. 160 |b. 3. 751b. 4. 1 ft. from stone. 
Page 122 
1. $11.20. 2. 68} min. 3. 63 mi. 4. 155 ft. 
5. 1), ft. 6. 561 mi. endo; 
Page 123 
1. 2 mo 2. 28, 3. 42 yd. 
Page 124 
1. $1210. 2.5da. 8. 27da. 4. $168.75. 5. 1172 sec. 
6. $864,000. 7. $1.05. 8. $750,000. 9. 7i da. 10. 8da., 6 da. 
Page 126 
1. 63 pt. 2. 248cu.in. 3. 1793 cu. in. 4. 22 in.; Sbint 
5. 22 sq. in. 6, 352} lb. Tam leas 8. 1.8,% in. 
Page 127 
1. $150, $225. 2. $2037, $582. 8. 455 ft., 1183 ft. 
4. $2073, $2764. 5. 361, 209. 6. 308. 7. 997.92, 3754.08. 
Page 128 
1.718; 2. 44. 3.1472 o Seals ab 6. 4.15. CO: 
8. $275, $750. 9. 28.8 sq. in. 10. 131.2 sq. in. 11. $9. 
127 12* AB: 13. $1225, $2450, $3675, $4900. 
14, $1000, $1500, $3500. 15. 12, 8, 12, 20, 28. 


GRAMMAR SCHOOL ALGEBRA 


Pages 117, 118 


ANSWERS 23 


Page 129 
Each root has the sign + on pages 129-132. 
teob: 2. 24. 3. 15. 40213 Sa. 2 7, 
Geek tyne len kat Sie 9. 19. 102° 32. 
Page 130 
Only a few types are given. 

2. 3+ 22, 3. 2a2?+4+ 1. Boo ts 
9. m+ 9. 12. p+ 5q. 14. 7—8y. 
Page 131 
Ler bi. 2. 61. 3. 63: 4. 71. 5. 79 
Page 132 
1. 8.97. 2. 44.1. 3. 1.05. 4. 0.343. d. 0.907. 
Ga0-5s, ToULie 671234. Orel: 10. 4607. 
11, 7008. 12. 9812. Logi 4) 14. 2.236. 15. 2.645. 

16. 2.828. Lieu LUO: 18. 33, 
Page 133 
ike) Soa 2. + 14. 3. 15 in. 4. +14. 
5. + 15. 6. + 12. fie SORRY 
Pages 134, 135 

1. +30. 2. + 40. Sen 25. 

4. 5 sec. 5. 10 ft., 15 ft. 62 20 1t,, 18 16. 
Ted, Se i4 it 21 in. 26 tt. 9. S4in., 70 ft., 252 in. 
10. 20. Lime 25: Py Bop se 

ish Se 14. + 89. 15. +81 

16. +110. 17. +90. 18. + 221 

1927-61: 20. 121. 21. §1 
22. 31. 23. 31 rd., 93 rd. 24. 142 in. 

25. 142 in., fin: 26. 100 rd., 25rd. 27. 12 in.,12.48 in. 
28. 7 in. 


a*b?(a — b) + ab(a + 5) 





(a + ¥)2(a — b) 


Page 136 





2 x?2y 3 4a+b 
“(@+y)(t-y? 40-28 


10. 


13. 


15. 


10. 


13. 


GRAMMAR SCHOOL ALGEBRA 


2 pqr —1 


” pg’r? — 1 


at + yt" 
uy? («+ Y) 
a2 + b2 





(a + b) (a — b)2 


. m2 -—m + 2. 


ab — a*b — ab? 





a2 — b2 


a3 — ab? + a2b — b3 — a3b — ab? 


ac — a2— b2 + be 
(b — e)(e —a) 























at — bt 
a> — 3.a2b — ab? + 63 





~ 


b2 — q2 
Water ah 
p(p+q+r) 
x 6 a2b + 2 63 
(a2 — b2)2 





Gea Cail 


eee (4 m3 — 5n?). 
2m 


. ab(a — Dd). 


m—1 
m+1 





11. 


we + y2z? g. wy + abx2y? - a?b? 
yz(w + x) ; xy? — a*b? 
82n + 12m 9 207 — 2xy 
" 9m? — 16 n2 "(a+ y)? 
sr+y4+3 1 _ bayz +a 
 g?— y2 " ab(x + y + 2) 
Page 137 
j oes 
2. 4 xy : 3. 9 3a 
x? — y? a? — 9 
5. 4@b gt 2mn 
b2 — at n2 — m2 
ry EL 
Viraie: + il 
37d — 27/2 aa 
10, Vay? + ty, 
1 — xty? 
12. Ee oo as 
(f=~1)? 
A ig 
4 a?b2 — 1 


82+ 6ayt+ y2— 822 — ry? 








16. 
2(3a + y)(2@ + y) 
Page 138 
m(m + 1) rey, 
n(n — 1) 
38xu-—y 6 2m+1 
r+3y 2 weal 
a? + 0 9 (m —1)(m +1)8 
a+b 4 
Ao 12. b—c. 
14. i : 
ab 
Page 139 
. 2ay (4ary + 8). 3. (a + b) (a? + 6?). 
e—l 6 “+ 4 
+3 ated 


10. 


13. 











ANSWERS 25 
x? + y? 
(eae to 8. (e+ 1)(4 +2). 9. ee 
a ae 2 
z : 11. m2n2?-—3mn—10. 12. oe) 
a+b-+ec x—y 
(xy — 1)? 14 a* + ab + B? 
a2 — y?2 . a2 + b2 
Pages 140-142 
2, 2, —b4 aud. 4. 8. 5. 10. Gal 
qgee.8. 8. 9, 22, 10.21, 11. 22=0..2=0. 
5. 13. 4. 14. 2. 15. 5. 16. 24. Lie sh 
1. 19. 4 20. 38a PE howe aces 23. 13. 
2. 25, —1, 26. —10. 27. 7 28. 7 peo ahs 
eames ree) Ao) go eames, 634.10. 
a 
ihe Nees 
PSG SUM Mase ee 6 10 382 4 
0 3 5 
Page 144 
2. 2 44 ae 4. 5,2. 
10, 1. 6. 4,5 Ce gale 8. 6, 1. 
Oa 1; 10. 8, 1. Tito 12. 20, 5. 
10, 9. [ance \Seeano vert 16g aa 
Page 145 
2, 3. 2. 4, 6. ae Goth 4. 8, 2. 
jay 62 1 Ch te ey eee as 
—d, —7 LOD LOB: lL. 12, -— 9. 12. 6, —8 
— 9,6 14.3: 30. 15. 32, 16. 16. 100, 50. 
Pages 146-149 
8, 11. Dy ey) Per hele eo eeY 
Pep e Bok, Gah ve<-6, 7.010, 10, 8. 26, — 30. 
58, 41 10. 4, 7. 1173 12. 3. 
t 14. $600, $400. 152 10,12: 16. 14, 16. 
<4 Re: 18. 18, 25. 19: 77, 33. 20. 36, 24, 
10, 24. 22. 4, 3. 23. 54, —50. 24. 30, 40. 
70, 80. 26. 63, — 4. 27. 4, 4. 28. 55, 6. 


35 


37 


38. 


. &+ 3800, + 800 
. 4% (4 + 1) = 9%, x = 8. 


GRAMMAR SCHOOL ALGEBRA 


3, 6. 30. 1, 3. 
—15,-—16. 934. 144, 64 
3, 2. 38. 2, 56 
Be RBy Py i, 
. $8-80+2 = 12(80+ 2), 2= 10. 
_ c+ 1B 2 ey 1848), 48: 


— _850 
Sa WO) 


oe Lif + 300), x=? 


Sli, 20%) eee SOG 
35,°0.7.\d. eens 6h Cire 
$95 755.60.) 940; se 
43. 7,7 
46. 6% (1 +2) =8%,2=? 


48, 
50. 


500. 
ie+1)=2+},0=? 


Pages 151, 152 


























GD ad? + be 3 
a 4 a? ; 
d—b b 
aac m—n — 26 
i paras ae) 10m 
1l+a+b—c dare 
teh ab—¢ 13. 1,2 
at—1 a—7 
a 2 avs 
6b ieee ab 16 
Aah Gee WR = DN ae 
D een g pean 19. 
2 2 
dm +067 0.— a 21 s+d 
“n+tac’ n+ac ‘ 2 ; 
22% 24 
ae Te 24. as 2.0Gh 25. 
a? a: 
p= Nae — 2 ab 22 28 
scnelee Ee Ge 32. 
ab 
Aes Ne AL cake t 34 
be—a? b?— a 
rae e(a be) c(ac — b”) 36 
a(ab — c2)’ b (ab — c2) 
10—7)b Dh 25.D 
“a(6c—5b) 6c—5B? 


St VPN a, a= +2 Vb? a2, 









































ae Aer 
a+e 
pie Ee ge 
m+n — Dp l—-m +n—p 
mtn bm—an 
11. oe 
a+b a+b 
14. Re ) 
2 2 
2 
eae ay AE 17. “i ’ zh 
a+la+l 
TOV WE TIRE og og 
n—1 ese 
s—d 29. ab+3b 
2 a 
c? — ac 26 3 be 
a a 
ne! leg a(t 
b=c ac—f 
eb hue b 
c(c — b) c(¢ — a) 
alas db) Bian) s 
Ne 26 
Dane gan 
39. + (6a + 5D). 


40. 
41. 


ANSWERS 
ab+ec 
ax = +(ab+c), «= +—— - 
13,6.  —«- 42..:«163, 88. 45. 21, 5. 
rer ene AL 46. 7, 8. 
bu —an bn—an 
Rage 154 
. —2, 1. 2. — 3, 2. 3. —4, 3 
wc A 8 Sh eS eae ay peeeney ye 
meeese saat eRe 1 ld = 65.2 
27) [4p 6 ee el be oad 
6, 6. 18. 5, 6. 19. —3, -—9 
wee DP), ea 235 30234 
11. 26. 4 yd.,7 yd. 


44, 


4. 
8. 
12. 
16. 
20. 
24. 
27, 134 ft., 6+ ft. 


27 


30 ct., 45 ct. 


ats 
Et iel: 
5, 4. 
Bel 
ol ee 
5, 7. 



























-URBANA 


C001 


ALGEBRA FOR BEGINNERS BOSTON 


3 0112 017080703 


UNIVERSITY OF ILLINOIS 


IANO UT 


512.9SM5A 





